Basic Numeracy Practice Questions — State PSC Exam — Page 14

Question 261Question →

A satellite transmission system encodes data packets using a cyclic redundancy check (CRC). The checksum value is determined by finding the remainder when the expression

T = 7 + 12 \times 53^{205}

is divided by

27

. What is the correct checksum value?

5

8

22

19

Show answer & explanation

Answer: 22

Answer

The correct checksum value is 22.

The correct answer is 22. To find the remainder, we apply modular arithmetic to the base: since

27 \times 2 = 54

, we know

53 \equiv -1 \pmod{27}

. Substituting this into the expression gives

7 + 12(-1)^{205}

. Because the exponent 205 is an odd number,

(-1)^{205} = -1

. Applying the order of operations, we multiply 12 by -1 to get -12, and then add 7, resulting in

-5

. Finally, to convert a negative remainder into a positive one, we add the divisor:

-5 + 27 = 22

.

Step-by-Step Solution

1

Identify the base and express it in modular arithmetic relative to the divisor 27.

53 \equiv -1 \pmod{27}

Simplifying the base to a small integer makes evaluating large exponents manageable. Since

54

is a multiple of

27

,

53

is exactly

1

less than a multiple.

2

Evaluate the exponentiated term using the simplified base.

53^{205} \equiv (-1)^{205} \equiv -1 \pmod{27}

Raising a negative one to an odd exponent results in negative one.

3

Substitute this result back into the expression and apply the proper order of operations (BODMAS).

7 + 12 \times (-1) = 7 - 12 = -5 \pmod{27}

According to mathematical rules, multiplication must be performed before addition.

4

Convert the negative remainder into a valid positive remainder.

-5 + 27 = 22

A standard remainder must be a non-negative integer less than the divisor. We achieve this by adding the divisor to the negative result.

Key Concept

Modular Arithmetic and Fermat's/Euler's Remainder Concepts

Estimated Time:1m 15s

Question 262Question →

A positive integer $X$ is of the form $X = 2^a \times 3^b$ , where $a$ and $b$ are positive integers. It is known that $X$ has exactly $15$ positive factors. If $X$ is a multiple of $9$ but is NOT a multiple of $27$ , which of the following is the exact value of $X$ ?

144

324

48

864

Show answer & explanation

Answer: 144

Answer

144

The correct value is determined by setting up the factor formula (a+1)(b+1) = 15. The only valid exponents that satisfy both the positive integer requirement and the divisibility constraints (b < 3 to avoid being a multiple of 27) are a=4 and b=2. Evaluating 2^4 \times 3^2 yields exactly 144.

Step-by-Step Solution

1

Identify the formula for the total number of positive factors.

For a prime factorization

X = p^a \times q^b

, the number of positive factors is

(a+1)(b+1)

.

This is a fundamental property of prime factorization.

2

Set up the equation based on the given number of factors.

(a+1)(b+1) = 15

.

The problem explicitly states that

X

has exactly 15 positive factors.

3

Determine the possible integer values for exponents

a

and

b

.

Since

a \ge 1

and

b \ge 1

, we need factor pairs of 15 that are

\ge 2

. The only pair is

3 \times 5

. This results in two possible cases:

(a+1=5, b+1=3)

which gives

a=4, b=2

; OR

(a+1=3, b+1=5)

which gives

a=2, b=4

.

Both

a

and

b

must be positive integers as defined in the question stem.

4

Apply the divisibility constraints to filter the two cases.

The condition 'NOT a multiple of 27' means

3^b

cannot be divisible by

3^3

, meaning

b

must be strictly less than

3

. Therefore,

b

must be

2

, which forces

a

to be

4

.

If

b=4

were chosen,

X

would be a multiple of

81

, which is a multiple of

27

, violating the negative constraint.

5

Calculate the final value of

X

.

X = 2^4 \times 3^2 = 16 \times 9 = 144

.

Substitute the verified exponents back into the original prime factorization.

Key Concept

Prime Factorization and Number of Factors

Estimated Time:1m 30s

Question 263Question →

Four identical delivery vans have different amounts of fuel remaining in their tanks. Arrange the vehicles based on their remaining fuel fraction from the least amount to the greatest amount.

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The correct ascending sequence of fuel remaining is Van R, Van P, Van S, and Van Q.

By converting all given amounts into decimals (

0.60

,

0.75

,

0.50

, and

0.65

), the values can be directly compared.

0.50

is the smallest and

0.75

is the largest, resulting in the correct ascending order of Van R, Van P, Van S, and Van Q.

Step-by-Step Solution

1

Convert all fractional fuel values into decimal form for straightforward comparison.

Van P (

\frac{3}{5}

) becomes 0.60, and Van R (

\frac{1}{2}

) becomes 0.50.

Converting all amounts to decimals provides a uniform format that is easier to logically order.

2

Order the decimal values from smallest to largest.

0.50 < 0.60 < 0.65 < 0.75

.

The question specifically requires arranging the vehicles from the least amount of fuel to the greatest amount.

Key Concept

Converting basic fractions to decimals to compare and order numerical values.

Question 264Question →

In an advanced error-checking algorithm, a data packet's integrity is verified by calculating a specific numerical signature. This signature is defined as the remainder when the value $V = 112^{2029} - 122^{2026}$ is divided by $13$ . What is the numerical signature of this data packet?

Show answer & explanation

Answer: 9

Answer

The numerical signature of the data packet is 9.

The expression evaluates to

-4 \pmod{13}

after simplifying the bases and exponents. Converting this to a valid positive remainder by adding the modulus

13

yields the correct answer,

9

.

Step-by-Step Solution

1

Reduce the base 112 modulo 13.

112 = 13 \times 8 + 8

, so

112 \equiv 8 \pmod{13}

.

In modular arithmetic, bases can be replaced by their remainders to simplify exponentiation.

2

Reduce the base 122 modulo 13.

122 = 13 \times 9 + 5

, so

122 \equiv 5 \pmod{13}

.

This simplifies the second term of the expression.

3

Apply Fermat's Little Theorem to simplify the exponent for the first term.

Since 13 is prime,

a^{12} \equiv 1 \pmod{13}

. For

8^{2029}

, dividing

2029

by

12

leaves a remainder of

1

(

2029 = 12 \times 169 + 1

). Thus,

8^{2029} \equiv 8^1 = 8 \pmod{13}

.

Large powers can be reduced by finding the remainder of the exponent divided by

p-1

.

4

Apply Fermat's Little Theorem to simplify the exponent for the second term.

For

5^{2026}

, dividing

2026

by

12

leaves a remainder of

10

(

2026 = 12 \times 168 + 10

). Thus,

5^{2026} \equiv 5^{10} \pmod{13}

.

This reduces the exponent to a manageable size.

5

Evaluate

5^{10} \pmod{13}

.

Since

5^2 = 25 \equiv -1 \pmod{13}

, we calculate

5^{10} = (5^2)^5 \equiv (-1)^5 = -1 \equiv 12 \pmod{13}

.

Using negative remainders for intermediate steps significantly speeds up manual exponentiation.

6

Subtract the evaluated terms and find the final positive remainder.

8 - 12 = -4

. To find the positive remainder, add the modulus 13:

-4 + 13 = 9

.

The final remainder must be a non-negative integer strictly less than the divisor.

Key Concept

Modular arithmetic, Fermat's Little Theorem, and handling negative remainders.

Alternative Method

Instead of using Fermat's Little Theorem (mod 12), you can find the exact order of the bases. Notice that

8^2 = 64 \equiv -1 \pmod{13}

, so

8^4 \equiv 1 \pmod{13}

. Similarly,

5^2 = 25 \equiv -1 \pmod{13}

, so

5^4 \equiv 1 \pmod{13}

. Both bases have an order of 4 modulo 13. Dividing the exponents by 4:

2029 \div 4

leaves a remainder of 1, so

8^{2029} \equiv 8^1 = 8

.

2026 \div 4

leaves a remainder of 2, so

5^{2026} \equiv 5^2 = 25 \equiv -1 \equiv 12

. The final subtraction is

8 - 12 = -4 \equiv 9 \pmod{13}

. This method requires less calculation.

Estimated Time:2m 0s

Question 265Question →

Analyze the given mathematical claims regarding the classification and properties of numbers:

I. If $p$ is a prime number strictly greater than $3$ , then $p^2 - 1$ is always a composite number divisible by $24$ .
II. The least common multiple (LCM) of the rational numbers $\frac{6}{5}$ and $\frac{12}{7}$ is a composite integer, while their highest common factor (HCF) is a non-integer rational number.
III. The numerical value of the expression $36 \div 6 \times 3$ belongs to the set of prime numbers.
IV. According to standard Euclidean division, the remainder when $-22$ is divided by $7$ is $-1$ , which is classified as a negative integer.

Which of the above claims are mathematically correct?

Only I, II, and III

Only I and II

Only I, II, and IV

Only I and IV

Show answer & explanation

Answer: Only I and II

Answer

The only mathematically correct claims are I and II.

The correct answer accurately identifies that only Claims I and II are valid. Claim I holds because the square of any prime

p>3

is congruent to

1 \pmod{24}

, making

p^2 - 1

a composite multiple of 24. Claim II is valid as the LCM of

6/5

and

12/7

evaluates precisely to the composite integer 12, and their HCF evaluates to

6/35

. Claims III and IV are mathematically unsound.

Step-by-Step Solution

1

Evaluate Claim I by testing the property of primes greater than 3.

Claim I is TRUE.

Any prime

p > 3

can be expressed as

6k \pm 1

. Thus,

p^2 - 1 = (6k \pm 1)^2 - 1 = 36k^2 \pm 12k = 12k(3k \pm 1)

. Since either

k

or

3k \pm 1

must be even, the expression is always a multiple of

12 \times 2 = 24

. Since

p \ge 5

,

p^2 - 1 \ge 24

, so it is always a composite number divisible by 24.

2

Evaluate Claim II by calculating the LCM and HCF of fractions

\frac{6}{5}

and

\frac{12}{7}

.

Claim II is TRUE.

LCM of fractions = LCM(numerators)/HCF(denominators) = LCM(6,12)/HCF(5,7) = 12/1 = 12, which is a composite integer. HCF of fractions = HCF(numerators)/LCM(denominators) = HCF(6,12)/LCM(5,7) = 6/35, which is a non-integer rational number.

3

Evaluate Claim III by applying the correct order of operations (BODMAS) to

36 \div 6 \times 3

.

Claim III is FALSE.

Division and multiplication have equal precedence and are evaluated from left to right.

36 \div 6 = 6

, and

6 \times 3 = 18

. The number 18 is composite, not prime.

4

Evaluate Claim IV by determining the correct Euclidean remainder of

-22 \div 7

.

Claim IV is FALSE.

In standard Euclidean division, the remainder

r

must satisfy

0 \le r < |divisor|

. For

-22

divided by

7

, the correct equation is

-22 = 7 \times (-4) + 6

. The remainder is 6 (a positive integer), not

-1

.

Key Concept

Classification of Numbers and Number Properties

Question 266Question →

What is the total number of positive divisors of the integer $10800$ that are perfect squares?

Show answer & explanation

Answer: 12

Answer

12

The correct answer is found by prime factorizing

10800

into

2^4 \times 3^3 \times 5^2

. A factor is a perfect square only when its prime exponents are all even integers. Counting the available even exponents from 0 up to the maximum exponent for each prime gives 3 choices for base 2 (0, 2, 4), 2 choices for base 3 (0, 2), and 2 choices for base 5 (0, 2). Multiplying these independent choices gives

3 \times 2 \times 2 = 12

.

Step-by-Step Solution

1

Find the prime factorization of

10800

.

10800 = 108 \times 100 = (2^2 \times 3^3) \times (2^2 \times 5^2) = 2^4 \times 3^3 \times 5^2

The prime factorization is necessary to systematically identify and count the divisors of the number based on exponent combinations.

2

Identify the mathematical condition for a divisor to be a perfect square.

A divisor

D = 2^a \times 3^b \times 5^c

is a perfect square if and only if the exponents

a

,

b

, and

c

are all even integers.

Perfect squares must have even exponents in their prime factorization so that their square roots are integers.

3

List the valid even exponent values for each prime base.

For base 2 (max exponent 4), valid even values are 0, 2, 4 (3 options). For base 3 (max exponent 3), valid even values are 0, 2 (2 options). For base 5 (max exponent 2), valid even values are 0, 2 (2 options).

These are the exponents that keep the divisor as a perfect square while still remaining a factor of

10800

.

4

Calculate the total number of perfect square divisors.

3 \times 2 \times 2 = 12

Applying the fundamental counting principle, we multiply the number of independent choices for each prime exponent.

Key Concept

Identifying perfect square factors using prime factorization

Estimated Time:1m 30s

Question 267Question →

What is the positive remainder when the expression $7 + 11 \times 24^{137}$ is divided by $25$ ?

21

4

7

18

Show answer & explanation

Answer: 21

Answer

21

The correct answer is found by substituting the base

24

with

-1

modulo

25

. Since

137

is an odd exponent,

(-1)^{137}

evaluates to

-1

. Following the order of operations, multiply this by

11

to get

-11

, and then add

7

to yield

-4

. Finally, to find the equivalent positive remainder modulo

25

, we add the divisor:

25 - 4 = 21

.

Step-by-Step Solution

1

Express the base

24

in terms of a negative remainder modulo

25

.

24 \equiv -1 \pmod{25}

Using

-1

simplifies the calculation of extremely large powers without fully expanding them.

2

Evaluate the exponent part modulo

25

.

24^{137} \equiv (-1)^{137} \equiv -1 \pmod{25}

An odd power of a negative number retains the negative sign.

3

Substitute this result back into the original expression and evaluate it.

7 + 11 \times (-1) = 7 - 11 = -4 \pmod{25}

Applying standard order of operations (BODMAS), multiplication must be performed before addition.

4

Convert the negative intermediate remainder into a positive remainder.

-4 + 25 = 21

A valid remainder in standard division must be non-negative and strictly less than the divisor.

Key Concept

Applying modular arithmetic properties, handling negative remainders appropriately, and maintaining the correct order of algebraic operations.

Question 268Question →

Consider the three fractions $X = \frac{16}{21}$ , $Y = \frac{8}{15}$ , and $Z = \frac{24}{35}$ . Based on the mathematical properties of factors and multiples for fractions, which of the following statements are correct? (Select all that apply)

Select all that apply

The Highest Common Factor (HCF) of the three fractions is

\frac{8}{105}

.

The Least Common Multiple (LCM) of the three fractions is

48

.

The Highest Common Factor (HCF) of the three fractions is

\frac{1}{48}

.

The Least Common Multiple (LCM) of the three fractions is

\frac{105}{8}

.

Show answer & explanation

Answer: The Highest Common Factor (HCF) of the three fractions is $\frac{8}{105}$ .; The Least Common Multiple (LCM) of the three fractions is $48$ .

Answer

The correct statements are that the HCF of the fractions is 8/105 and the LCM of the fractions is 48.

The mathematical rules for fractions state that the HCF is determined by the ratio of the HCF of the numerators to the LCM of the denominators, which evaluates to 8/105. Similarly, the LCM is determined by the ratio of the LCM of the numerators to the HCF of the denominators, which evaluates to 48/1 or 48. The statements identifying these two specific values correctly represent these properties.

Step-by-Step Solution

1

Verify that all given fractions are in their simplest form.

The fractions 16/21, 8/15, and 24/35 cannot be reduced further because their respective numerators and denominators have no common prime factors.

The standard formulas for finding the HCF and LCM of fractions strictly require all fractions to be reduced to their lowest terms first.

2

Extract the numerators and calculate their HCF and LCM.

Numerators are 16, 8, and 24. Their HCF is 8. Their LCM is 48.

These values are the necessary numerator components for the overarching fraction formulas.

3

Extract the denominators and calculate their HCF and LCM.

Denominators are 21, 15, and 35. Their HCF is 1 (since 3x7, 3x5, and 5x7 share no common prime factor across all three). Their LCM is 105 (3 x 5 x 7).

These values are the necessary denominator components for the overarching fraction formulas.

4

Apply the rule for the Highest Common Factor (HCF) of fractions.

HCF = HCF of Numerators / LCM of Denominators = 8 / 105.

This establishes the truth value for the statements regarding the HCF.

5

Apply the rule for the Least Common Multiple (LCM) of fractions.

LCM = LCM of Numerators / HCF of Denominators = 48 / 1 = 48.

This establishes the truth value for the statements regarding the LCM.

Key Concept

Calculating Highest Common Factor (HCF) and Least Common Multiple (LCM) for Fractions

Question 269Question →

When a positive integer $x$ is divided by $17$ , the remainder is $12$ . When another positive integer $y$ is divided by $17$ , the remainder is $15$ . What is the positive remainder when the expression $4x - 5y$ is divided by $17$ ?

4

7

10

16

Show answer & explanation

Answer: 7

Answer

7

Using the properties of modular arithmetic, we substitute the remainders into the expression:

4(12) - 5(15) = 48 - 75 = -27

. To find the equivalent positive remainder modulo

17

, we add the nearest larger multiple of

17

, which is

34

. Thus,

-27 + 34 = 7

.

Step-by-Step Solution

1

Express the given properties using modular arithmetic.

x \equiv 12 \pmod{17}

and

y \equiv 15 \pmod{17}

This allows us to substitute the remainders directly into the expression.

2

Substitute the remainders into the expression

4x - 5y

.

4(12) - 5(15)

By the properties of modular arithmetic, the remainder of a sum or difference is the sum or difference of the individual remainders.

3

Evaluate the arithmetic expression.

48 - 75 = -27

Simplifying the multiplication and subtraction gives the intermediate negative remainder.

4

Convert the negative remainder into a valid positive remainder.

-27 \equiv -27 + 2(17) \equiv -27 + 34 \equiv 7 \pmod{17}

A valid remainder must be a positive integer between

0

and the divisor minus

1

.

Key Concept

Modular Arithmetic Operations and Negative Remainders

Question 270Question →

An architect is planning to pave a rectangular courtyard using exactly $240$ identical square stones. To ensure the space is functional, both the length and the width of the courtyard must consist of at least $5$ stones. Assuming the stones cannot be cut, how many different unordered pairs of dimensions (length and width) are possible for this courtyard?

16

10

6

20

Show answer & explanation

Answer: 6

Answer

6

The correct answer is derived by finding the total number of factor pairs for 240 (which is 10 pairs) and subtracting the 4 pairs that contain a factor smaller than 5 (1, 2, 3, and 4). This leaves exactly 6 valid unordered pairs.

Step-by-Step Solution

1

Find the prime factorization of 240.

240 = 2^4 \times 3^1 \times 5^1

Prime factorization is required to calculate the total number of factors (divisors) of the number.

2

Calculate the total number of factors for 240.

Total factors =

(4+1)(1+1)(1+1) = 5 \times 2 \times 2 = 20

.

Adding 1 to each exponent in the prime factorization and multiplying them yields the total number of divisors.

3

Calculate the total number of unordered pairs of dimensions.

Total pairs =

20 \div 2 = 10

.

Since 240 is not a perfect square, its 20 factors form exactly 10 distinct pairs

(a, b)

such that

a \times b = 240

.

4

Identify and exclude pairs where at least one dimension is less than 5.

The factors less than 5 are 1, 2, 3, and 4. These correspond to 4 invalid pairs: (1, 240), (2, 120), (3, 80), and (4, 60).

The problem specifically states that both dimensions must consist of at least 5 stones.

5

Calculate the final number of valid pairs.

10 - 4 = 6

valid pairs.

Subtracting the 4 invalid pairs from the total 10 pairs leaves the number of pairs that satisfy all conditions.

Key Concept

Factors, Multiples, and Prime Factorization

Alternative Method

Instead of finding all factors using prime factorization, one could systematically list the factor pairs starting from

1 \times 240

up to the square root of

240

(which is approximately

15.4

). The pairs are (1,240), (2,120), (3,80), (4,60), (5,48), (6,40), (8,30), (10,24), (12,20), and (15,16). Eliminating the first four pairs that contain a number smaller than 5 leaves exactly 6 valid options.

Estimated Time:1m 30s

Question 271Question →

In a chemical manufacturing plant, three automated valves release specific additives into a continuous mixing tank. Valve A opens every $\frac{9}{2}$ minutes, Valve B opens every $\frac{15}{4}$ minutes, and Valve C opens every $\frac{45}{8}$ minutes. The system is programmed to record a 'synchronization event' whenever all three valves open at the exact same instant.

If the mixing process runs continuously for exactly $12$ hours, how many synchronization events will be recorded during this period? (Assume a synchronization event is recorded at the very beginning of the process, which counts as the first event).

129

32

271

33

Show answer & explanation

Answer: $33$

Answer

The correct answer is

33

synchronization events.

The synchronization interval is the LCM of the fractions

\frac{9}{2}

,

\frac{15}{4}

, and

\frac{45}{8}

. Using the formula

\frac{\text{LCM of numerators}}{\text{HCF of denominators}}

, we get

\frac{\text{LCM}(9, 15, 45)}{\text{HCF}(2, 4, 8)} = \frac{45}{2} = 22.5

minutes. Over a

12

-hour period (

720

minutes), there are precisely

\frac{720}{22.5} = 32

complete intervals. Since a synchronization event is recorded at the very beginning of the process, the total number of recorded events is

32 + 1 = 33

.

Step-by-Step Solution

1

Identify the mathematical concept required to find the synchronization interval.

The shortest time interval for all valves to open together is the Least Common Multiple (LCM) of their individual time cycles.

Synchronization occurs at common multiples of the cycle times, and the next event always occurs at the least common multiple.

2

Calculate the LCM of the fractions

\frac{9}{2}

,

\frac{15}{4}

, and

\frac{45}{8}

.

\text{LCM of fractions} = \frac{\text{LCM}(9, 15, 45)}{\text{HCF}(2, 4, 8)}

.

The standard formula for the LCM of fractions requires finding the LCM of the numerators divided by the HCF of the denominators.

3

Compute the LCM of numerators and HCF of denominators.

\text{LCM}(9, 15, 45) = 45

and

\text{HCF}(2, 4, 8) = 2

. Thus, the synchronization interval is

\frac{45}{2} = 22.5

minutes.

This establishes that every

22.5

minutes, all three valves will open simultaneously.

4

Calculate the total time in minutes for the

12

-hour shift.

12 \text{ hours} \times 60 \text{ minutes/hour} = 720 \text{ minutes}

.

Units must be consistent to accurately determine the number of intervals.

5

Determine the number of full intervals and total synchronization events.

\text{Number of intervals} = \frac{720}{22.5} = 32

. Total events

= 32 + 1 = 33

.

The division yields the number of periods between events. We add

1

to account for the initial synchronization event explicitly mentioned in the problem (the fencepost principle).

Key Concept

Calculating the Least Common Multiple (LCM) of fractions and applying it to a temporal boundary condition (fencepost counting).

Estimated Time:2m 0s

Question 272Question →

An industrial control system tracks part serial numbers using a cyclic validation algorithm. A batch identifier $N$ is generated using the formula $N = 65^{202} - 32^{145} - 45$ . The system validates this batch by computing the positive remainder when $N$ is divided by the base modulus $33$ . What is the expected positive remainder for this batch?

Show answer & explanation

Answer: 23

Answer

The expected positive remainder is 23.

By applying the principles of modular arithmetic, we can simplify each term of the expression

65^{202} - 32^{145} - 45

modulo

33

. First,

65 \equiv -1 \pmod{33}

, so

65^{202} \equiv (-1)^{202} = 1

. Next,

32 \equiv -1 \pmod{33}

, so

32^{145} \equiv (-1)^{145} = -1

. Substituting these simplified values back into the expression yields

1 - (-1) - 45 = 2 - 45 = -43

. To find the positive remainder when dividing by

33

, we add a multiple of

33

(in this case,

66

) to

-43

, which results in

23

.

Step-by-Step Solution

1

Express the base numbers

65

and

32

modulo

33

.

65 \equiv -1 \pmod{33}

and

32 \equiv -1 \pmod{33}

Finding remainders close to

1

or

-1

significantly simplifies the evaluation of large exponents.

2

Evaluate the first term,

65^{202}

modulo

33

.

(-1)^{202} = 1

An even power of a negative one results in positive one.

3

Evaluate the second term,

32^{145}

modulo

33

.

(-1)^{145} = -1

An odd power of a negative one results in negative one.

4

Substitute the simplified terms back into the original expression modulo

33

.

N \equiv 1 - (-1) - 45 \equiv 2 - 45 \equiv -43 \pmod{33}

Combines all parts of the expression to find the overall remainder before final normalization.

5

Convert the negative remainder into a valid positive remainder.

-43 \equiv -43 + 66 = 23 \pmod{33}

Remainders must be positive integers between

0

and the modulus minus one. Adding the smallest multiple of

33

that is greater than

43

(which is

66

) gives the valid positive remainder.

Key Concept

Modular Arithmetic and Remainder Theorem with Negative Remainders

Alternative Method

You could also simplify the constant term first:

-45 \equiv -12 \equiv 21 \pmod{33}

. Then the expression becomes

1 - (-1) + 21 = 23

.

Estimated Time:1m 30s

Question 273Question →

A municipal election committee is distributing ballots to various polling stations. When they pack the ballots in bundles of $45$ , $54$ , or $72$ , they find that they are always left with $38$ , $47$ , and $65$ unbundled ballots, respectively. If the total number of ballots printed is the largest possible 4-digit number satisfying these conditions, what is the exact total number of ballots?

Show answer & explanation

Answer: 9713

Answer

9713

The problem describes a scenario where the difference between each divisor (

45

,

54

,

72

) and its respective remainder (

38

,

47

,

65

) is exactly

7

. This means that if

7

more ballots were added, the total would be perfectly divisible by all three numbers. Therefore, the required total is exactly

7

less than a common multiple of these divisors. The LCM of

45

,

54

, and

72

is

1080

. The largest

4

-digit multiple of

1080

is

9720

(

1080 \times 9

). Subtracting the constant difference of

7

from

9720

gives the final answer of

9713

.

Step-by-Step Solution

1

Calculate the difference between each bundle size and its corresponding remainder.

45 - 38 = 7

,

54 - 47 = 7

, and

72 - 65 = 7

.

To identify if there is a constant difference, which allows the use of the LCM minus constant method.

2

Determine the Least Common Multiple (LCM) of the bundle sizes

45

,

54

, and

72

.

\text{LCM}(45, 54, 72) = 1080

.

The LCM represents the smallest bundle size that perfectly divides by all three numbers. Prime factorizations:

45 = 3^2 \times 5

,

54 = 2 \times 3^3

,

72 = 2^3 \times 3^2

. LCM

= 2^3 \times 3^3 \times 5 = 1080

.

3

Find the largest 4-digit multiple of the LCM.

1080 \times 9 = 9720

.

The problem asks for the largest 4-digit number. Dividing

9999

by

1080

yields

9.258...

, so the largest integer multiplier is

9

.

4

Subtract the constant difference from this largest multiple.

9720 - 7 = 9713

.

Since each division left a remainder that was

7

short of a full bundle, subtracting

7

from a perfect multiple satisfies all three remainder conditions.

Key Concept

Solving simultaneous remainder problems where the difference between divisors and remainders is constant, by utilizing the LCM and scaling to a specific boundary range.

Question 274Question →

Consider the following set of 8 numbers:

S = \left\{ -7, 0, \pi, \frac{22}{7}, \sqrt{169}, \sqrt{50}, 2.5, 0.\overline{45} \right\}

Let:
$P =$ the number of integers in $S$
$Q =$ the number of rational numbers in $S$
$R =$ the number of irrational numbers in $S$
$T =$ the number of whole numbers in $S$

Calculate the exact value of $(Q \times R) + (P \times T)$ .

Show answer & explanation

Answer: 18

Answer

18

Based on mathematical definitions: P (integers) = 3 because -7, 0, and 13 are integers. Q (rationals) = 6 because it includes the 3 integers plus 22/7, 2.5, and 0.4545... R (irrationals) = 2 because it includes pi and sqrt(50). T (whole numbers) = 2 because it includes the non-negative integers 0 and 13. Plugging these into (Q * R) + (P * T) yields (6 * 2) + (3 * 2) = 18.

Step-by-Step Solution

1

Evaluate and simplify each number in the set to determine its properties.

\sqrt{169} simplifies to exactly 13. The repeating decimal 0.\overline{45} can be written as a fraction (45/99).

Numbers must be evaluated in their simplest form to avoid misclassification.

2

Determine the value of P by counting the integers.

The integers are -7, 0, and 13. Therefore, P = 3.

Integers include positive numbers, negative numbers, and zero, with no decimal or fractional parts.

3

Determine the value of Q by counting the rational numbers.

The rational numbers are -7, 0, 22/7, 13, 2.5, and 0.\overline{45}. Therefore, Q = 6.

Rational numbers are any numbers that can be expressed as a ratio of two integers.

4

Determine the value of R by counting the irrational numbers.

The irrational numbers are \pi and \sqrt{50}. Therefore, R = 2.

Irrational numbers have non-terminating, non-repeating decimal expansions.

5

Determine the value of T by counting the whole numbers.

The whole numbers are 0 and 13. Therefore, T = 2.

Whole numbers consist only of the non-negative integers.

6

Substitute the counts into the requested mathematical expression and calculate the final result.

(6 \times 2) + (3 \times 2) = 12 + 6 = 18.

To answer the specific question given in the stem.

Key Concept

Classification of real numbers into integers, rational numbers, irrational numbers, and whole numbers.

Question 275Question →

A manufacturing plant produces a total of $10 + 35 \times 41^{43}$ small electronic components. These components are to be distributed equally into shipping containers that hold exactly $42$ units each. How many components will be left unpacked after filling as many containers as possible?

25

39

17

3

Show answer & explanation

Answer: 17

Answer

17

To find the leftover components, we must calculate the remainder of the total amount modulo 42. Using modular arithmetic, 41 is congruent to -1 modulo 42. Raising -1 to the odd power of 43 yields -1. Multiplying this by 35 gives -35. Adding 10 results in -25. Because a physical remainder must be positive, we add the divisor (42) to the intermediate value of -25 to find the equivalent positive remainder, which is 17.

Step-by-Step Solution

1

Set up the modulo equation to find the remainder of the total components divided by 42.

(10 + 35 \times 41^{43}) \pmod{42}

The number of unpacked components represents the mathematical remainder when the total quantity is divided by the container capacity.

2

Simplify the large base (41) with respect to the divisor (42).

41 \equiv -1 \pmod{42}

Replacing a large base with its smallest absolute congruent value simplifies large exponent calculations.

3

Substitute -1 into the expression and evaluate the odd exponent.

10 + 35 \times (-1)^{43} \equiv 10 + 35 \times (-1) \pmod{42}

A negative base raised to an odd power evaluates to a negative result.

4

Perform the multiplication and addition following standard order of operations.

10 - 35 = -25 \pmod{42}

Multiplication must be completed strictly before addition.

5

Convert the negative remainder into a valid strictly positive remainder.

-25 + 42 = 17

Physical remainders cannot be negative. To find the equivalent positive remainder in modular arithmetic, add the divisor to the negative result.

Key Concept

Modular Arithmetic and Negative Remainders

Estimated Time:1m 30s

Question 276Question →

A civil supplies department procures three batches of edible oil weighing $0.\overline{6}$ metric tonnes, $1.\overline{3}$ metric tonnes, and $2.\overline{2}$ metric tonnes. The department intends to repackage the entire procured quantity into smaller, identical standardized containers. To minimize the number of containers used without mixing the batches, what should be the maximum capacity of each standardized container?

6.\overline{6}

metric tonnes

0.\overline{2}

metric tonnes

0.\overline{6}

metric tonnes

0.1

metric tonnes

Show answer & explanation

Answer: $0.\overline{2}$ metric tonnes

Answer

The maximum capacity of each standardized container should be

0.\overline{2}

metric tonnes.

The maximum capacity of the standardized containers is determined by finding the Highest Common Factor (HCF) of the oil volumes. First, the given recurring decimals are converted to fractions:

\frac{2}{3}

,

\frac{4}{3}

, and

\frac{20}{9}

. The HCF of these fractions is found by taking the HCF of the numerators (2, 4, 20), which is 2, and dividing it by the LCM of the denominators (3, 3, 9), which is 9. This calculation yields

\frac{2}{9}

, which is equivalent to the recurring decimal

0.\overline{2}

.

Step-by-Step Solution

1

Convert the recurring decimals representing the batch weights into fractions in their simplest form.

0.\overline{6} = \frac{6}{9} = \frac{2}{3}

,

1.\overline{3} = \frac{13-1}{9} = \frac{12}{9} = \frac{4}{3}

, and

2.\overline{2} = \frac{22-2}{9} = \frac{20}{9}

.

Mathematical operations involving divisors and multiples are most accurate when recurring decimals are expressed exactly as fractions.

2

Identify the mathematical requirement to find the maximum standardized container capacity.

We must determine the Highest Common Factor (HCF) of the three fractional quantities.

To divide all batches completely into identical containers of maximum capacity without leftovers, the container size must be the greatest common divisor of all the given volumes.

3

Apply the standard formula for calculating the HCF of multiple fractions.

\text{HCF of fractions} = \frac{\text{HCF of numerators}}{\text{LCM of denominators}}

.

This formula ensures that the resulting fraction will evenly divide each of the original fractional quantities.

4

Calculate the HCF of the numerators and the LCM of the denominators for

\frac{2}{3}

,

\frac{4}{3}

, and

\frac{20}{9}

.

Numerators (2, 4, 20): The

\text{HCF}

is 2. Denominators (3, 3, 9): The

\text{LCM}

is 9.

These values provide the numerator and denominator for the final maximum capacity.

5

Form the final fraction and convert it back to a recurring decimal to match the given options.

\text{Maximum capacity} = \frac{2}{9} = 0.\overline{2}

metric tonnes.

Converting back to recurring decimal format answers the problem in the original units.

Key Concept

Fractions and Decimals - Converting Recurring Decimals and Finding HCF of Fractions

Question 277Question →

Two positive integers $A$ and $B$ (both strictly greater than $12$ ) have a Highest Common Factor (HCF) of $12$ and a Least Common Multiple (LCM) of $504$ . Based on this information, which of the following statements are true?

Select all that apply

The sum of the integers

A

and

B

can be

156

.

The positive difference between

A

and

B

can be

132

.

There are exactly four distinct unordered pairs

\{A, B\}

that satisfy all given conditions.

The Least Common Multiple (LCM) of the fractions

\frac{1}{A}

and

\frac{1}{B}

is

\frac{1}{504}

.

Show answer & explanation

Answer: The sum of the integers $A$ and $B$ can be $156$ .; The positive difference between $A$ and $B$ can be $132$ .

Answer

The statements confirming that the sum of the integers can be 156 and that their positive difference can be 132 are both correct.

Based on the prime factorization constraints, the valid unordered pairs

\{A, B\}

are

\{24, 252\}

,

\{36, 168\}

, and

\{72, 84\}

. Because

72 + 84 = 156

and

168 - 36 = 132

, the descriptive statements asserting that the sum can be

156

and the positive difference can be

132

are logically true.

Step-by-Step Solution

1

Use the property that the product of two numbers equals the product of their HCF and LCM.

A \times B = 12 \times 504 = 6048

.

This establishes the fundamental numerical relationship between the two integers.

2

Express

A

and

B

in terms of their HCF to simplify.

Let

A = 12x

and

B = 12y

, where

x

and

y

are coprime integers. Substituting into the product gives

144xy = 6048

, which simplifies to

xy = 42

.

Factoring out the HCF ensures that any pairs found will inherently respect the greatest common divisor requirement.

3

Find all coprime pairs

(x, y)

that multiply to 42 and calculate the corresponding integer pairs

\{A, B\}

.

The coprime pairs for

42

are

(1, 42)

,

(2, 21)

,

(3, 14)

, and

(6, 7)

. The corresponding

\{A, B\}

pairs are

\{12, 504\}

,

\{24, 252\}

,

\{36, 168\}

, and

\{72, 84\}

.

These pairs represent all mathematical possibilities before applying the specific contextual constraints.

4

Apply the problem's contextual condition that both integers must be strictly greater than 12.

The pair

\{12, 504\}

must be excluded. The valid pairs are exactly three:

\{24, 252\}

,

\{36, 168\}

, and

\{72, 84\}

.

Filtering out invalid mathematical solutions ensures strict adherence to the stated facts.

5

Evaluate the given statements against the three valid integer pairs.

The sum

72 + 84 = 156

validates the first statement. The difference

168 - 36 = 132

validates the second. The count of valid pairs is

3

, refuting the third statement. The LCM of

1/A

and

1/B

is

1/12

, refuting the fourth.

Systematically verifying which descriptive statements accurately reflect the discovered pairs.

Key Concept

Applying HCF and LCM properties alongside algebraic constraints to evaluate paired integers.

Estimated Time:2m 0s

Question 278Question →

A jeweler is cutting equal-length pieces of gold wire from three different spools to make uniform necklace links without any wastage. The three spools contain gold wire of lengths $\frac{27}{2}$ cm, $\frac{45}{4}$ cm, and $\frac{63}{5}$ cm. What is the maximum possible length of each uniform gold wire piece that can be cut? (Provide your answer as a precise decimal)

Show answer & explanation

Answer: 0.45

Answer

0.45

The maximum uniform length is found by computing the Highest Common Factor (HCF) of the fractional wire lengths. Applying the formula yields an HCF of numerators (9) over the LCM of denominators (20), giving

\frac{9}{20}

, which correctly converts to exactly

0.45

cm.

Step-by-Step Solution

1

Determine the mathematical operation required for the scenario.

Calculate the Highest Common Factor (HCF) of the three fractional lengths.

Cutting multiple lengths into the maximum possible equal segments without wastage is defined by the HCF.

2

Apply the rule for finding the HCF of fractions.

HCF = HCF(Numerators) / LCM(Denominators)

This is the standard formula for finding the greatest common divisor of multiple fractional values.

3

Find the HCF of the numerators 27, 45, and 63.

HCF(27, 45, 63) = 9

9 is the largest integer that divides perfectly into 27, 45, and 63.

4

Find the LCM of the denominators 2, 4, and 5.

LCM(2, 4, 5) = 20

20 is the smallest integer that is a multiple of 2, 4, and 5.

5

Combine the results into the final fraction and convert to a decimal.

\frac{9}{20} = 0.45

cm

The question requires the precise decimal representation of the fraction.

Key Concept

Calculating the Highest Common Factor (HCF) of fractions to solve optimization word problems.

Question 279Question →

A botanical garden is installing a new irrigation system and has three main supply hoses measuring $\frac{44}{5}$ meters, $\frac{77}{4}$ meters, and $\frac{121}{10}$ meters in length. The landscaping team needs to cut all three hoses into smaller, equal-length segments to connect to individual planters. If no material can be wasted, what is the minimum total number of segments that can be produced from all three hoses combined?

Show answer & explanation

Answer: 73

Answer

73 segments

The minimum number of total segments is achieved when each segment is cut to its maximum possible equal length. This maximum length is the HCF of the three fractions (44/5, 77/4, 121/10), which evaluates to 11/20 meters. Dividing the original lengths by this HCF yields exactly 16, 35, and 22 pieces respectively, totaling 73 individual segments.

Step-by-Step Solution

1

Determine the mathematical requirement for minimizing the number of segments.

Identify that the Highest Common Factor (HCF) of the hose lengths must be found.

To get the minimum number of pieces, each piece must be of the maximum possible equal length.

2

Apply the rule for finding the HCF of fractions.

Use the formula:

\text{HCF} = \frac{\text{HCF of numerators}}{\text{LCM of denominators}}

.

This formula allows the calculation of the greatest common divisor for non-integer fractional values.

3

Calculate the HCF of the numerators.

The numerators are

44

,

77

, and

121

. Their HCF is

11

.

The largest integer that perfectly divides

44

,

77

, and

121

is

11

(

44 = 11 \times 4

,

77 = 11 \times 7

,

121 = 11 \times 11

).

4

Calculate the LCM of the denominators.

The denominators are

5

,

4

, and

10

. Their LCM is

20

.

The smallest integer that is a multiple of

5

,

4

, and

10

is

20

.

5

Determine the maximum segment length.

The segment length is

\frac{11}{20}

meters.

Combining the results from the previous steps using the fraction HCF formula yields the length.

6

Calculate the number of segments produced from each hose.

First hose:

\frac{44}{5} \div \frac{11}{20} = 16

. Second hose:

\frac{77}{4} \div \frac{11}{20} = 35

. Third hose:

\frac{121}{10} \div \frac{11}{20} = 22

.

Dividing the total length of each hose by the length of one segment gives the segment count per hose.

7

Sum the segment counts.

16 + 35 + 22 = 73

segments.

The question asks for the minimum total number of segments produced from all three hoses combined.

Key Concept

Calculating the Highest Common Factor (HCF) of fractions and applying it to optimize division in real-world scenarios.

Estimated Time:2m 0s

Question 280Question →

A cryptography algorithm generates a numerical verification key, $V$ , through a specific two-step mathematical process. First, it determines the value of $K$ , which is strictly defined as the positive remainder when $-26$ is divided by $7$ . Second, it calculates the final key using the expression $V = K \times (18 \div 6 \times 3)$ . Based on the fundamental classification of numbers, which of the following accurately describes the final value of $V$ ?

It is a positive, even, and composite number.

It is a negative, odd integer.

It is a positive, even, and prime number.

It is a positive, odd, and composite number.

Show answer & explanation

Answer: It is a positive, even, and composite number.

Answer

The final value is 18, which is uniquely classified as a positive, even, and composite number.

By applying the formal definition of the modulo operation, the positive remainder of

-26

divided by

7

is found by the equation

-26 = 7 \times (-4) + 2

, giving

K = 2

. Next, using the correct order of operations (left to right for consecutive division and multiplication), the expression evaluates to

(18 \div 6) \times 3 = 3 \times 3 = 9

. Multiplying these results yields

V = 2 \times 9 = 18

. The number 18 is classified as positive (greater than zero), even (divisible by 2), and composite (having factors other than 1 and itself).

Step-by-Step Solution

1

Determine the positive remainder when

-26

is divided by

7

to find the value of

K

.

K = 2

By the formal definition of Euclidean division, the remainder must be non-negative. We must express

-26

as

7 \times (-4) + 2

, which gives a valid positive remainder of

2

.

2

Evaluate the mathematical expression

18 \div 6 \times 3

following the correct order of operations.

The expression evaluates to

9

.

According to the BODMAS rule, division and multiplication share the same level of precedence and must be evaluated strictly from left to right. Therefore,

(18 \div 6) = 3

, and

3 \times 3 = 9

.

3

Calculate the final value of the verification key

V

and classify it.

V = 2 \times 9 = 18

. The number

18

is a positive, even, and composite number.

The number

18

is greater than zero (positive), perfectly divisible by

2

(even), and possesses divisors other than

1

and itself, such as

2, 3,

and

6

(composite).

Key Concept

Classification of Numbers and Fundamental Arithmetic Operations

Basic Numeracy

Answer

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Common Mistakes

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Common Mistakes

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