Basic Numeracy Practice Questions — State PSC Exam — Page 6

Question 101Question →

A city's public transport network features three distinct tram lines that operate on continuous circular routes departing from a central station. Tram Line 1 completes its route every $\frac{45}{4}$ minutes. Tram Line 2 completes its route every $\frac{25}{2}$ minutes, and Tram Line 3 takes $\frac{75}{8}$ minutes per loop. If all three trams depart from the central station simultaneously, how many minutes will it take for them to depart together again for the first time?

Show answer & explanation

Answer: 112.5

Answer

It will take 112.5 minutes for all three trams to depart together again.

The correct answer is found by taking the Least Common Multiple of the fractional times. By finding the LCM of the numerators (225) and dividing it by the Highest Common Factor of the denominators (2), we get 225/2, which evaluates to exactly 112.5 minutes.

Step-by-Step Solution

1

Determine the mathematical operation required to find when the events will synchronize.

Identify the need to calculate the Least Common Multiple (LCM) of the fractions

\frac{45}{4}

,

\frac{25}{2}

, and

\frac{75}{8}

.

The trams will meet again at a time that is a common multiple of their individual loop durations. The 'first time' indicates the least common multiple is needed.

2

Apply the rule for calculating the LCM of fractional values.

Use the formula:

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

.

To synchronize fractional frequencies, the numerators must reach a common multiple while strictly maintaining the largest common baseline unit defined by the denominators.

3

Calculate the LCM of the numerators: 45, 25, and 75.

The LCM of 45, 25, and 75 is 225.

Prime factorization:

45 = 3^2 \times 5

;

25 = 5^2

;

75 = 3 \times 5^2

. Taking the highest powers gives

3^2 \times 5^2 = 9 \times 25 = 225

.

4

Calculate the HCF of the denominators: 4, 2, and 8.

The HCF of 4, 2, and 8 is 2.

2 is the largest integer that can divide 4, 2, and 8 without leaving a remainder.

5

Compute the final synchronized time.

Divide the LCM of numerators by the HCF of denominators:

\frac{225}{2} = 112.5

.

Applying the values to the fraction LCM formula yields the exact time in minutes.

Key Concept

Calculating the Least Common Multiple (LCM) for fractions to solve simultaneous event problems.

Question 102Question →

Consider the numerical expression $E = 8^{52} - 3^{27}$ . If $u$ represents the unit digit of the positive integer $E$ , which of the following correctly classifies the number $u$ ?

It is an odd composite number.

It is an even composite number.

It is a negative integer.

It is a unique number that is neither prime nor composite.

Show answer & explanation

Answer: It is an odd composite number.

Answer

The calculated unit digit is 9, which is classified as an odd composite number.

Evaluating the expression requires finding the unit digits of both exponential terms. The unit digit of

8^{52}

is 6, and the unit digit of

3^{27}

is 7. When subtracting 7 from 6 in a larger positive number, borrowing from the tens place results in

16 - 7 = 9

. The number 9 is odd and has three distinct factors (1, 3, 9), making it an odd composite number.

Step-by-Step Solution

1

Determine the unit digit of

8^{52}

.

The unit digit is 6.

The cyclicity pattern for powers of 8 is 4 (ending in 8, 4, 2, 6). Since 52 is a perfect multiple of 4 (remainder is 0), the unit digit matches the 4th power in the cycle, which is 6.

2

Determine the unit digit of

3^{27}

.

The unit digit is 7.

The cyclicity pattern for powers of 3 is 4 (ending in 3, 9, 7, 1). Dividing the exponent 27 by 4 leaves a remainder of 3. Therefore, the unit digit matches the 3rd power in the cycle (

3^3

), which ends in 7.

3

Calculate the unit digit

u

of the difference

E

.

u = 9

Subtracting the unit digits gives

6 - 7

. Because

E

is a positive integer, we must borrow 10 from the next higher place value in the base-10 system, yielding

16 - 7 = 9

.

4

Classify the resulting number 9.

9 is an odd composite number.

The integer 9 cannot be evenly divided by 2 (making it odd) and has positive divisors other than 1 and itself (1, 3, and 9), which classifies it as composite.

Key Concept

Integration of power cyclicity rules and fundamental number classification.

Question 103Question →

A metal fabrication company has three long copper rods measuring $\frac{54}{5}$ meters, $\frac{81}{10}$ meters, and $\frac{108}{25}$ meters. The company needs to cut all three rods into smaller segments of equal length such that no copper material is left over. To minimize the total number of segments, what is the maximum possible length of each individual segment?

\frac{162}{25}

meters

\frac{324}{5}

meters

\frac{27}{50}

meters

\frac{27}{5}

meters

Show answer & explanation

Answer: $\frac{27}{50}$ meters

Answer

The maximum possible length of each segment is 27/50 meters.

The maximum possible length for the equal segments is found by calculating the HCF of the given fractions. Using the mathematical rule for fractions, HCF = HCF(numerators) / LCM(denominators). The HCF of 54, 81, and 108 is 27. The LCM of 5, 10, and 25 is 50. Therefore, the maximum length is 27/50 meters.

Step-by-Step Solution

1

Identify the mathematical operation required to find the maximum possible equal length.

The problem requires finding the Highest Common Factor (HCF) of the three fractional lengths.

The rods must be cut into equal pieces without wastage, and the length of each piece must be maximized to minimize the number of segments.

2

Recall the formula for finding the HCF of fractions.

HCF of fractions = (HCF of numerators) / (LCM of denominators).

This formula is necessary to accurately compute the greatest common divisor of non-integer values.

3

Calculate the HCF of the numerators: 54, 81, and 108.

The factors give 54 = 2 × 27, 81 = 3 × 27, and 108 = 4 × 27. The HCF is 27.

We need the greatest integer that divides all the numerators evenly.

4

Calculate the LCM of the denominators: 5, 10, and 25.

The multiples of 25 are 25, 50, 75, etc. Since 50 is divisible by both 5 and 10, the LCM is 50.

We need the smallest common multiple for the denominators to complete the fraction formula.

5

Construct the final fraction.

The maximum length is 27/50 meters.

Dividing the computed numerator HCF by the denominator LCM gives the correct value.

Key Concept

HCF of Fractions

Question 104Question →

A numerical analysis task requires evaluating four specific values to classify them into their correct number sets. The values are defined as follows:

- $K = 18 - 6 \div 2 + 1$
- $L =$ The unit digit of $8^{32}$
- $M =$ The remainder when $-23$ is divided by $5$
- $N =$ The Highest Common Factor (HCF) of $\frac{3}{4}$ and $\frac{9}{10}$

Based on the correct mathematical evaluation of these expressions, which of the following statements regarding their classification are mathematically correct?

Select all that apply

The value of

K

is a perfect square, and

M

is the only prime number among the integer results.

The value of

N

is a rational fraction strictly between

0

and

1

, and

L

is an even composite number.

The value of

K

is a prime number, and

N

is an improper fraction greater than

1

.

The value of

L

is a positive odd integer, and

M

is a negative integer.

Show answer & explanation

Answer: The value of $K$ is a perfect square, and $M$ is the only prime number among the integer results.; The value of $N$ is a rational fraction strictly between $0$ and $1$ , and $L$ is an even composite number.

Answer

The mathematically correct statements are that K is a perfect square, M is the only prime among the integer results, N is a rational fraction between 0 and 1, and L is an even composite number.

Based on rigorous mathematical evaluation, the true values are

K = 16

,

L = 6

,

M = 2

, and

N = \frac{3}{20}

.

K

(

16

) is a perfect square, and

M

(

2

) is the only prime among the integer results (

16, 6, 2

). Furthermore,

N

(

0.15

) is a rational fraction strictly between

0

and

1

, and

L

(

6

) is an even composite number. Therefore, these descriptive classifications perfectly match the evaluated properties.

Step-by-Step Solution

1

Evaluate expression K using proper BODMAS rules.

K = 16

Division must be performed before addition and subtraction.

K = 18 - (6 \div 2) + 1 = 18 - 3 + 1 = 16

, which classifies as a perfect square.

2

Determine the unit digit of L based on cyclicity.

L = 6

The unit digit of powers of

8

follows a 4-step cycle (

8, 4, 2, 6

). Since the exponent

32

is perfectly divisible by

4

, the unit digit is the 4th in the cycle, which is

6

(an even composite number).

3

Calculate the mathematically correct positive remainder for M.

M = 2

By the formal division algorithm,

-23 = 5 \times (-5) + 2

. Remainder must be non-negative, so the remainder is

2

, which is an even prime number.

4

Compute the HCF of the given fractions for N.

N = \frac{3}{20}

The HCF of fractions is computed as

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

.

\text{HCF}(3,9) = 3

and

\text{LCM}(4,10) = 20

. Thus

N = \frac{3}{20} = 0.15

, a rational number strictly between

0

and

1

.

5

Cross-reference the correctly evaluated numbers against the provided statements.

The statements categorizing

K

as a perfect square,

M

as the only prime among integers,

N

between

0

and

1

, and

L

as an even composite are correct.

The integer results are

16, 6

, and

2

, where exactly one (

2

) is prime. The other statements rely on distinct computational and conceptual errors.

Key Concept

Applying fundamental arithmetic rules and modular arithmetic to properly classify numbers into distinct mathematical sets.

Estimated Time:1m 30s

Question 105Question →

An event organizer is arranging chairs for a large conference. When the chairs are arranged in rows of $18$ , $24$ , or $36$ , there are always exactly $5$ chairs left over. However, when the chairs are arranged in rows of $13$ , all chairs are perfectly accommodated with none left over. What is the minimum possible total number of chairs the organizer has?

Show answer & explanation

Answer: 221

Answer

The minimum possible total number of chairs is 221.

The correct answer is derived by first establishing that any number leaving a remainder of

5

when divided by

18

,

24

, and

36

must be of the form

72k + 5

, where

72

is the LCM of the divisors. By systematically checking values of

k

, we find that

k=3

is the smallest integer that makes the expression

(72k + 5)

perfectly divisible by

13

, resulting in

72(3) + 5 = 221

.

Step-by-Step Solution

1

Set up the conditions for the total number of chairs mathematically.

Let the total number of chairs be

N

. We are given

N \equiv 5 \pmod{18}

,

N \equiv 5 \pmod{24}

,

N \equiv 5 \pmod{36}

, and

N \equiv 0 \pmod{13}

.

Translating the word problem into modular arithmetic helps systematically apply the concepts of LCM and divisibility.

2

Find the Least Common Multiple (LCM) of the first set of divisors.

The divisors are

18

,

24

, and

36

. Their prime factorizations are

18 = 2 \times 3^2

,

24 = 2^3 \times 3

, and

36 = 2^2 \times 3^2

. The LCM is

2^3 \times 3^2 = 8 \times 9 = 72

.

Any number that leaves the same remainder when divided by multiple divisors must be a multiple of their LCM plus that remainder.

3

Express

N

using the LCM and the common remainder.

Since

N

leaves a remainder of

5

when divided by

18

,

24

, or

36

, we can write

N = 72k + 5

, where

k

is a non-negative integer (

k = 0, 1, 2, \dots

).

This general formula captures all possible numbers of chairs that satisfy the first condition.

4

Apply the final divisibility condition to find

k

.

We require

N

to be perfectly divisible by

13

, meaning

72k + 5 \equiv 0 \pmod{13}

.

This guarantees the solution satisfies the second condition where arranging chairs in rows of 13 leaves no remainder.

5

Simplify the congruence modulo 13 and solve for

k

.

Divide

72

by

13

to find the remainder:

72 = 13 \times 5 + 7

. So,

72k \equiv 7k \pmod{13}

. The equation becomes

7k + 5 \equiv 0 \pmod{13}

. Testing values for

k

: if

k=1

,

7(1)+5=12

(not divisible); if

k=2

,

7(2)+5=19

(not divisible); if

k=3

,

7(3)+5=26

(divisible by

13

, since

26 = 13 \times 2

). The smallest valid

k

is

3

.

Finding the smallest non-negative integer

k

ensures we find the minimum possible number of chairs.

6

Calculate the final value of

N

.

N = 72(3) + 5 = 216 + 5 = 221

.

Substituting

k=3

back into our general formula gives the final answer.

Key Concept

Solving simultaneous remainder and divisibility conditions using the Least Common Multiple (LCM).

Question 106Question →

A gourmet coffee roaster has three batches of rare coffee beans weighing $\frac{15}{2}$ kg, $\frac{25}{4}$ kg, and $\frac{35}{8}$ kg respectively. The roaster wants to package all the beans into smaller, equal-sized bags such that each bag contains the maximum possible weight of coffee beans without mixing the batches, and no beans are left over. What should be the weight of each bag?

\frac{525}{2}

kg

\frac{525}{8}

kg

\frac{5}{8}

kg

\frac{5}{2}

kg

Show answer & explanation

Answer: $\frac{5}{8}$ kg

Answer

The weight of each bag should be

\frac{5}{8}

kg.

The problem asks for the maximum possible equal capacity for the bags, which requires calculating the Highest Common Factor (HCF) of the given fractional weights. The HCF of a set of fractions is found by calculating the HCF of their numerators (

15

,

25

,

35

) and dividing it by the LCM of their denominators (

2

,

4

,

8

). The HCF of the numerators is

5

, and the LCM of the denominators is

8

. Therefore, the correct weight is

\frac{5}{8}

kg.

Step-by-Step Solution

1

Determine the mathematical operation required.

We need to find the Highest Common Factor (HCF) of the given fractional weights.

The bags must be of equal size, hold the maximum possible weight, and leave no remainder, which matches the definition of HCF.

2

Recall the formula for finding the HCF of fractions.

The formula is: (HCF of numerators) / (LCM of denominators).

This formula ensures the resulting fraction correctly divides all the given fractions into integers.

3

Calculate the HCF of the numerators.

The numerators are

15

,

25

, and

35

. Their HCF is

5

.

5

is the largest integer that divides

15

,

25

, and

35

without leaving a remainder.

4

Calculate the LCM of the denominators.

The denominators are

2

,

4

, and

8

. Their LCM is

8

.

8

is the smallest integer that is a multiple of

2

,

4

, and

8

.

5

Combine the results to find the HCF of the fractions.

The final HCF is

\frac{5}{8}

.

Substituting the calculated HCF and LCM into the fraction formula yields

\frac{5}{8}

.

Key Concept

Finding the Highest Common Factor (HCF) of fractions.

Question 107Question →

Three mathematical values, $X$ , $Y$ , and $Z$ , are defined below:

- $X = \frac{22}{7} - \pi$
- $Y = \text{The positive remainder obtained when } -23 \text{ is divided by } 6$
- $Z = 2 + 18 \div 3 \times 2$

Based on the fundamental properties of numbers, which of the following statements correctly classifies all three values?

Value

X

is an irrational number,

Y

is neither prime nor composite, and

Z

is an even composite number.

Value

X

is an irrational number,

Y

is neither prime nor composite, and

Z

is an odd prime number.

Value

X

is an irrational number,

Y

is an odd prime number, and

Z

is an even composite number.

Value

X

is a rational number,

Y

is neither prime nor composite, and

Z

is an even composite number.

Show answer & explanation

Answer: Value $X$ is an irrational number, $Y$ is neither prime nor composite, and $Z$ is an even composite number.

Answer

Value

X

is an irrational number,

Y

is neither prime nor composite, and

Z

is an even composite number.

First,

X

is the difference between a rational number (

22/7

) and an irrational number (

\pi

), which results in a non-zero irrational number. Second, applying the division algorithm (

a = bq + r

where

0 \le r < b

), we evaluate

-23 = 6 \times (-4) + 1

. Thus, the positive remainder

Y

is 1, which is uniquely defined as a natural number that is neither prime nor composite. Finally, evaluating

Z

requires applying BODMAS rules from left to right for multiplication and division:

18 \div 3 = 6

, then

6 \times 2 = 12

, and

2 + 12 = 14

. The result 14 is an even composite number.

Step-by-Step Solution

1

Evaluate and classify the value of

X = \frac{22}{7} - \pi

.

X

is a non-zero irrational number.

Since

22/7

is a rational approximation and not strictly equal to the irrational number

\pi

, their difference yields a non-zero irrational number.

2

Determine the true positive remainder of

-23 \div 6

to find

Y

.

Y = 1

, which is neither prime nor composite.

Using the division algorithm (

a = bq + r

where

0 \le r < b

), we write

-23 = 6 \times (-4) + 1

. The positive remainder is 1, a natural number defined as neither prime nor composite.

3

Calculate

Z = 2 + 18 \div 3 \times 2

using proper operational precedence.

Z = 14

, which is an even composite number.

According to BODMAS, division and multiplication are evaluated strictly from left to right before addition. Thus,

18 \div 3 = 6

, followed by

6 \times 2 = 12

, and finally

2 + 12 = 14

.

Key Concept

Classification of Real Numbers, Modulo Arithmetic, and Order of Operations

Question 108Question →

An arithmetic expression is formulated as $7 + 10 \times 28^{67}$ . When this entire value is divided by $29$ , what is the resulting positive remainder?

3

12

17

26

Show answer & explanation

Answer: 26

Answer

26

By applying the property of modular arithmetic, we know that

28 \equiv -1 \pmod{29}

. Substituting this into the expression simplifies the exponent:

28^{67} \equiv (-1)^{67} \pmod{29}

. Because 67 is an odd integer,

(-1)^{67} = -1

. The expression becomes

7 + 10 \times (-1)

. Following the order of operations, the multiplication is resolved first to yield

-10

, and then the addition gives

7 - 10 = -3

. Finally, to find the equivalent positive remainder modulo 29, the divisor is added to the negative result:

29 - 3 = 26

.

Step-by-Step Solution

1

Express the base

28

in terms of modulo

29

.

28 \equiv -1 \pmod{29}

Converting to a small negative base drastically simplifies the calculation of large exponents.

2

Evaluate the exponent term modulo

29

.

28^{67} \equiv (-1)^{67} = -1 \pmod{29}

An odd power of

-1

evaluates to

-1

.

3

Apply the standard order of operations (BODMAS) to the expression.

7 + 10 \times (-1) = 7 - 10 = -3 \pmod{29}

Multiplication must be performed before addition.

4

Convert the negative remainder into a valid positive remainder.

29 + (-3) = 26

Standard positive remainders must be non-negative and strictly less than the divisor, which is achieved by adding the divisor to the negative result.

Key Concept

Modular arithmetic principles, specifically managing negative bases and converting negative remainders, applied alongside the standard order of operations.

Alternative Method

One could utilize Fermat's Little Theorem, which states

a^{p-1} \equiv 1 \pmod{p}

for prime

p

. Here,

28^{28} \equiv 1 \pmod{29}

. The power 67 can be broken down:

28^{67} = (28^{28})^2 \times 28^{11} \equiv 1^2 \times (-1)^{11} = -1 \pmod{29}

. This rigorous path mathematically verifies the simpler direct substitution of

-1

.

Estimated Time:1m 0s

Question 109Question →

Evaluate the arithmetic expression $17 - 5 \times 6$ . What is the positive remainder when the result of this expression is divided by $8$ ?

0

3

5

7

Show answer & explanation

Answer: 3

Answer

The positive remainder is 3.

Following the correct order of operations, multiplication must be performed before subtraction. The expression becomes 17 - 30, which equals -13. To find the positive remainder when a negative number is divided by 8, we add the smallest multiple of 8 that is greater than 13 (which is 16) to the negative number. Thus, -13 + 16 = 3. The correct remainder is 3.

Step-by-Step Solution

1

Apply the correct order of operations to evaluate the multiplication first.

17 - (5 \times 6) = 17 - 30

According to mathematical order of operations, multiplication takes precedence over subtraction.

2

Complete the subtraction to find the final integer value.

17 - 30 = -13

Subtracting a larger positive number from a smaller one results in a negative integer.

3

Determine the positive remainder when

-13

is divided by

8

.

-13 = 8 \times (-2) + 3

To find a valid positive remainder for a negative number, add the divisor until the result is positive:

-13 + 8 = -5

, and

-5 + 8 = 3

.

Key Concept

Order of Operations and Negative Remainders

Estimated Time:45s

Question 110Question →

Consider the arithmetic expression $E = 18^{41} - 22^{60} \times 37^{42}$ . When the value of $E$ is divided by $10$ , what is the resulting positive remainder?

8

4

6

9

Show answer & explanation

Answer: 4

Answer

4

Evaluating the remainders of each term modulo 10 (which is equivalent to finding their unit digits), we obtain 8, 6, and 9 respectively. Applying the correct order of operations, multiplication precedes subtraction:

8 - (6 \times 9) = 8 - 54 = -46

. To convert -46 into a valid positive remainder modulo 10, we add the nearest larger multiple of 10 (which is 50), yielding

-46 + 50 = 4

.

Step-by-Step Solution

1

Find the unit digit (remainder modulo 10) of

18^{41}

.

8

The unit digit cycle of 8 is (8, 4, 2, 6). Since 41 divided by 4 leaves a remainder of 1, the unit digit corresponds to the first value in the cycle, which is

8^1 = 8

.

2

Find the unit digit (remainder modulo 10) of

22^{60}

.

6

The unit digit cycle of 2 is (2, 4, 8, 6). Since 60 is perfectly divisible by 4 (remainder 0), we use the 4th value in the cycle, which is

2^4 = 16 \rightarrow 6

.

3

Find the unit digit (remainder modulo 10) of

37^{42}

.

9

The unit digit cycle of 7 is (7, 9, 3, 1). Since 42 divided by 4 leaves a remainder of 2, the unit digit corresponds to the second value in the cycle, which is

7^2 = 49 \rightarrow 9

.

4

Substitute these values into the expression following the order of operations.

8 - 6 \times 9 = 8 - 54 = -46

According to the BODMAS rule, multiplication must be performed before subtraction.

5

Convert the negative result to a valid positive remainder modulo 10.

4

To convert a negative remainder to a positive one, add a multiple of the divisor that brings the value above zero.

-46 + 50 = 4

.

Key Concept

Unit digit cyclicity, modular arithmetic, and the conversion of negative remainders.

Question 111Question →

Read the following three statements concerning the properties of numbers:

1. The integer $0$ is considered a positive even number.
2. The fraction $\frac{22}{7}$ is a rational number, whereas the constant $\pi$ is an irrational number.
3. The product of any two irrational numbers always results in an irrational number.

Which of the statements given above is/are mathematically correct?

2 only

1 and 2 only

2 and 3 only

1, 2, and 3

Show answer & explanation

Answer: 2 only

Answer

Only the second statement is mathematically accurate: the fraction 22/7 is rational, and pi is irrational.

The correct answer is '2 only' because it is the only mathematically sound statement. The number 0 is neither positive nor negative, invalidating statement 1. The product of two irrational numbers can be rational (such as multiplying two identical square roots), invalidating statement 3. Statement 2 correctly identifies that any ratio of integers (like 22/7) is rational, while the mathematical constant pi is irrational.

Step-by-Step Solution

1

Evaluate Statement 1: 'The integer 0 is considered a positive even number.'

Statement 1 is incorrect.

By mathematical definition, the integer 0 is an even number (since it is divisible by 2 with no remainder), but it serves as the boundary between positive and negative numbers. It is strictly neutral, neither positive nor negative.

2

Evaluate Statement 2: 'The fraction 22/7 is a rational number, whereas the constant pi is an irrational number.'

Statement 2 is correct.

A rational number is any number that can be expressed as a fraction p/q where p and q are integers and q is not 0. Since 22 and 7 are integers, 22/7 is definitely rational. The mathematical constant pi represents a non-terminating, non-repeating decimal and cannot be expressed exactly as a simple fraction, making it irrational.

3

Evaluate Statement 3: 'The product of any two irrational numbers always results in an irrational number.'

Statement 3 is incorrect.

The set of irrational numbers is not closed under multiplication. For example, multiplying the irrational number sqrt(2) by another irrational number sqrt(2) yields 2, which is a rational integer.

4

Synthesize the evaluations to find the correct choice.

Since only statement 2 is mathematically valid, the correct option is '2 only'.

Matching our findings with the provided options confirms the final answer.

Key Concept

Classification of Real Numbers: Properties of Rational and Irrational Numbers

Question 112Question →

When the mathematical expression $44^{63} - 17^2$ is divided by $45$ , what is the final positive remainder?

Show answer & explanation

Answer: 25

Answer

25

Applying modular arithmetic rules, the base

44

is congruent to

-1 \pmod{45}

. Raising

-1

to an odd power (

63

) keeps the value as

-1

. For the second term,

17^2

equals

289

, which leaves a remainder of

19

when divided by

45

. Subtracting the second remainder from the first gives

-1 - 19 = -20

. Because a standard remainder must be positive, adding the divisor (

45

) to

-20

yields the final correct answer of

25

.

Step-by-Step Solution

1

Apply modular arithmetic to the first term,

44^{63}

, relative to the divisor

45

.

44 \equiv -1 \pmod{45}

, so

44^{63} \equiv (-1)^{63} = -1 \pmod{45}

Using negative remainders for bases close to the divisor significantly simplifies large power calculations.

2

Evaluate the second term,

17^2

, and find its remainder when divided by

45

.

17^2 = 289

. Dividing

289

by

45

yields a quotient of

6

(

270

) with a remainder of

19

.

The constant term must be reduced modulo 45 to properly combine it with the first term.

3

Combine the simplified terms according to the original expression structure.

-1 - 19 = -20 \pmod{45}

The remainder of a difference is equivalent to the difference of the individual remainders.

4

Convert the resulting negative remainder into an equivalent positive remainder.

-20 + 45 = 25

Standard remainders must be non-negative. Adding the divisor to a negative remainder provides the mathematically correct positive value.

Key Concept

Modular Arithmetic and Negative Remainders

Question 113Question →

Which of the following statements regarding the classification and fundamental properties of numbers are mathematically correct? (Select all that apply)

Select all that apply

The sum of any rational number and any irrational number always results in an irrational number.

The integer

0

is considered a positive number because it sits to the right of negative values on the number line.

The mathematical constant

\pi

is exactly equal to the fraction

\frac{22}{7}

, meaning it is classified as a rational number.

The number

2

is the only even prime number, while all even integers greater than

2

are composite numbers.

Show answer & explanation

Answer: The sum of any rational number and any irrational number always results in an irrational number.; The number $2$ is the only even prime number, while all even integers greater than $2$ are composite numbers.

Answer

The correct statements are that the sum of a rational and an irrational number is always irrational, and that the number 2 is the only even prime number.

The correct statements highlight foundational number properties: a rational number added to an irrational number always forms a new irrational number, and the number 2 is unique as the only even prime number in mathematics, with all other even numbers above it containing multiple divisors.

Step-by-Step Solution

1

Evaluate the closure property of rational and irrational numbers described in the first statement.

The statement is determined to be mathematically true, as combining a rational with an irrational always yields an irrational result.

To verify if the first classification rule provided is accurate.

2

Analyze the sign classification of the integer 0 as presented in the second statement.

The statement is found to be false because 0 is a neutral integer, neither positive nor negative.

To test the common misconception that 0 holds a positive value.

3

Differentiate between the exact value of the constant pi and its fractional approximations in the third statement.

The statement is deemed false because pi is fundamentally irrational, and 22/7 is only an approximation.

To address the frequent conflation of pi with its common rational substitute.

4

Determine the prime and composite nature of even integers as claimed in the fourth statement.

The statement is recognized as true since 2 is the solely existing even prime number, with all larger even integers being composite.

To confirm the parity properties of prime numbers.

Key Concept

Classification of rational vs irrational numbers, neutral integers, and prime properties.

Question 114Question →

What is the Highest Common Factor (HCF) of the fractions $\frac{2}{3}$ , $\frac{4}{9}$ , and $\frac{8}{15}$ ?

\frac{8}{3}

\frac{2}{45}

\frac{2}{3}

\frac{8}{45}

Show answer & explanation

Answer: $\frac{2}{45}$

Answer

The correct answer is

\frac{2}{45}

.

The HCF of a set of fractions is found by dividing the HCF of the numerators by the LCM of the denominators. The numerators are 2, 4, and 8, and their HCF is 2. The denominators are 3, 9, and 15, and their LCM is 45. Combining these yields the correct value of

\frac{2}{45}

.

Step-by-Step Solution

1

Identify the formula for finding the HCF of fractions.

HCF of fractions = (HCF of numerators) / (LCM of denominators).

This standard formula is required to correctly solve for the highest common factor of any set of fractions.

2

Extract the numerators and find their Highest Common Factor (HCF).

The numerators are 2, 4, and 8. Their HCF is 2.

2 is the largest integer that divides 2, 4, and 8 without leaving a remainder.

3

Extract the denominators and find their Least Common Multiple (LCM).

The denominators are 3, 9, and 15. Their LCM is 45.

45 is the smallest positive integer that is a multiple of 3, 9, and 15.

4

Apply the calculated values to the fraction HCF formula.

The final fraction is

\frac{2}{45}

.

Substituting the HCF of the numerators (2) and the LCM of the denominators (45) into the formula gives the final answer.

Key Concept

The HCF of two or more fractions is calculated by dividing the HCF of their numerators by the LCM of their denominators.

Question 115Question →

Calculate the true positive remainder obtained upon dividing the numerical expression $67^{95} - 3 \times 53^{42}$ by $17$ .

Show answer & explanation

Answer: 4

Answer

The correct positive remainder is 4.

By evaluating the expression under modulo

17

, we first simplify the base numbers:

67 \equiv -1 \pmod{17}

and

53 \equiv 2 \pmod{17}

. This reduces the overarching expression to

(-1)^{95} - 3 \times 2^{42}

. The first term trivially becomes

-1

. For the second term, we can utilize the fact that

2^4 = 16 \equiv -1 \pmod{17}

. Therefore,

2^{42}

can be broken down into

(2^4)^{10} \times 2^2 \equiv (-1)^{10} \times 4 \equiv 4 \pmod{17}

. Substituting these simplified values back into the expression yields

-1 - 3(4) = -13

. Because standard remainders must be positive, we adjust the negative result by adding the divisor

17

to

-13

, which gives the true positive remainder of

4

.

Step-by-Step Solution

1

Reduce the base numbers

67

and

53

to smaller equivalent values modulo

17

.

67 = 17 \times 4 - 1 \implies 67 \equiv -1 \pmod{17}

. And

53 = 17 \times 3 + 2 \implies 53 \equiv 2 \pmod{17}

.

To drastically simplify large exponentiations by substituting smaller, manageable equivalent bases.

2

Evaluate the remainder of the first term,

67^{95} \pmod{17}

.

(-1)^{95} = -1 \pmod{17}

.

An odd exponent applied to a base of

-1

preserves the negative sign.

3

Simplify the second term's exponentiation,

2^{42} \pmod{17}

, by identifying a nearby power of

2

that relates to

17

.

Observe that

2^4 = 16 \equiv -1 \pmod{17}

.

Finding a power that equals

1

or

-1

modulo

17

creates a highly efficient shortcut for reducing massive exponents.

4

Break down

2^{42}

using the established property of

2^4

.

2^{42} = (2^4)^{10} \times 2^2 \equiv (-1)^{10} \times 4 \equiv 1 \times 4 = 4 \pmod{17}

.

To substitute the

-1

equivalence and systematically compute the modular value of the second term.

5

Combine both simplified terms into the original arithmetic expression.

The expression evaluates to

-1 - 3 \times 4 = -1 - 12 = -13 \pmod{17}

.

To find the overall aggregate modular value of the complete mathematical expression.

6

Convert the negative result into the equivalent true positive remainder.

-13 + 17 = 4

.

By definition, a valid remainder must be a non-negative integer strictly less than the divisor.

Key Concept

Modular Arithmetic, Exponent Rules, and Negative Remainders

Estimated Time:3m 0s

Question 116Question →

In the real number system, numbers are broadly classified into rational and irrational categories. Based on strict mathematical definitions, which of the following is an irrational number?

\frac{22}{7}

\sqrt{25}

3.14159

\pi

Show answer & explanation

Answer: $\pi$

Answer

The mathematical constant

\pi

is an irrational number.

The constant

\pi

cannot be expressed exactly as a fraction of two integers. Its decimal representation is non-terminating and non-repeating, which is the defining characteristic of an irrational number.

Step-by-Step Solution

1

Recall the definition of a rational number.

A rational number can be written exactly as a fraction

\frac{p}{q}

where

p

and

q

are integers and

q \neq 0

.

Establishing the rule allows us to eliminate options that fit this criteria.

2

Evaluate the fractional and decimal options.

\frac{22}{7}

is explicitly a ratio of integers.

3.14159

is a terminating decimal, which can also be written as a fraction.

Both perfectly fit the definition of a rational number and must be eliminated.

3

Evaluate the square root option.

\sqrt{25}

simplifies to the integer

5

.

Since

5 = \frac{5}{1}

, it is also a rational number.

4

Evaluate the mathematical constant.

\pi

represents a non-terminating, non-repeating decimal that cannot be written exactly as a fraction.

Numbers with non-terminating and non-repeating decimal expansions are classified as irrational numbers.

Key Concept

Distinguishing between rational numbers (exact fractions, terminating decimals, perfect roots) and irrational numbers (non-terminating, non-repeating decimals like the constant pi).

Question 117Question →

Let $N = 53^{91} - 31^{82}$ . When the value of $N$ is divided by $11$ , which of the following represents the correct remainder?

5

6

2

0

Show answer & explanation

Answer: 5

Answer

The correct remainder is 5.

By finding the remainders of the bases (

53 \equiv -2 \pmod{11}

and

31 \equiv -2 \pmod{11}

) and reducing the exponents using Fermat's Little Theorem modulo 10 (

91 \equiv 1 \pmod{10}

and

82 \equiv 2 \pmod{10}

), the expression simplifies to

(-2)^1 - (-2)^2 = -6

. Converting this negative remainder to a true positive remainder yields

-6 + 11 = 5

.

Step-by-Step Solution

1

Determine the remainders of the bases 53 and 31 when divided by 11.

53 \equiv 9 \equiv -2 \pmod{11}

and

31 \equiv 9 \equiv -2 \pmod{11}

.

Converting to negative remainders significantly simplifies the calculations for large powers.

2

Apply Fermat's Little Theorem to reduce the exponents.

Since 11 is a prime number, the cyclicity is

11 - 1 = 10

. Reducing the exponents modulo 10 gives

91 \equiv 1 \pmod{10}

and

82 \equiv 2 \pmod{10}

.

Fermat's theorem states that

a^{p-1} \equiv 1 \pmod{p}

, meaning exponents can be reduced by finding their remainder when divided by

p-1

.

3

Evaluate the reduced expression under modulo 11.

(-2)^1 - (-2)^2 = -2 - 4 = -6

.

Substituting the simplified bases and exponents provides the intermediate remainder.

4

Convert the negative intermediate remainder into a valid positive remainder.

-6 + 11 = 5

.

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

Key Concept

Applying Fermat's Little Theorem and correctly handling negative remainders in modular arithmetic.

Question 118Question →

In a basic encryption algorithm, a numerical value $V$ is generated using the formula $V = 47^{91} - 19^{3}$ . The final security key is determined by the positive remainder when $V$ is divided by $12$ . What is the value of the final security key?

8

6

2

4

Show answer & explanation

Answer: 4

Answer

4

The correct answer is found by applying modular arithmetic rules to reduce each part of the expression modulo 12. First,

47 \equiv -1 \pmod{12}

, so

47^{91} \equiv (-1)^{91} = -1

. Second,

19 \equiv 7 \pmod{12}

, and since

7^2 = 49 \equiv 1 \pmod{12}

, we get

7^3 \equiv 7 \times 1 = 7

. Subtracting these yields

-1 - 7 = -8

. To find the equivalent positive remainder, we add the divisor:

-8 + 12 = 4

.

Step-by-Step Solution

1

Find the remainder of

47^{91}

modulo 12.

Since

47 = 12 \times 4 - 1

, it is easier to use

47 \equiv -1 \pmod{12}

. Therefore,

47^{91} \equiv (-1)^{91} = -1 \pmod{12}

.

Using negative remainders for numbers close to a multiple of the divisor simplifies large exponent calculations.

2

Find the remainder of

19^{3}

modulo 12.

Since

19 = 12 \times 1 + 7

, we have

19 \equiv 7 \pmod{12}

. Thus,

19^3 \equiv 7^3 \pmod{12}

. We know

7^2 = 49 \equiv 1 \pmod{12}

. Multiplying by 7 gives

7^3 \equiv 1 \times 7 = 7 \pmod{12}

.

Reducing the base before exponentiation makes the calculation manageable without needing to calculate

19^3

completely.

3

Subtract the individual remainders.

(-1) - 7 = -8 \pmod{12}

.

Applying modular arithmetic properties to combine the terms in the original expression.

4

Convert the negative remainder to a positive remainder.

-8 + 12 = 4

.

The standard remainder definition requires a non-negative integer strictly less than the divisor.

Key Concept

Modular arithmetic, negative remainders, and power reduction rules.

Question 119Question →

Consider the exponential equation:

4^x - 3 \cdot 2^{x+2} + 32 = 0

Determine the sum of all real values of $x$ that satisfy this equation.

Show answer & explanation

Answer: 5

Answer

5

By applying the laws of indices, the original expression transforms into a quadratic equation in terms of

2^x

. Solving

y^2 - 12y + 32 = 0

yields

y=4

and

y=8

, which correspond precisely to

x=2

and

x=3

. Their sum is 5.

Step-by-Step Solution

1

Express all terms with a common base of 2.

4^x

becomes

(2^x)^2

and

2^{x+2}

becomes

4 \cdot 2^x

.

Creating a common base allows the equation to be transformed into a standard polynomial form.

2

Rewrite the equation using the new terms.

(2^x)^2 - 12(2^x) + 32 = 0

Simplifying the coefficients makes it easier to spot the quadratic structure.

3

Perform a substitution to solve the quadratic equation.

Letting

y = 2^x

gives

y^2 - 12y + 32 = 0

. Factoring yields

(y - 4)(y - 8) = 0

, so

y = 4

or

y = 8

.

Substitution converts a complex exponential equation into a simple quadratic one.

4

Solve for the original variable

x

.

2^x = 4 \implies x = 2

, and

2^x = 8 \implies x = 3

.

The question asks for the values of

x

, not the intermediate substitution variable

y

.

5

Calculate the sum of all valid

x

values.

2 + 3 = 5

This addresses the specific final requirement of the question stem.

Key Concept

Solving exponential equations reducible to quadratics using index laws.

Estimated Time:1m 30s

Question 120Question →

A synchronized scheduling system operates on a repeating 19-millisecond cycle. A specific event is triggered at a timestamp $T$ in milliseconds, given by the formula $T = 37^{45} - 17 \times 40^{23}$ . To find the exact position within the current cycle when the event occurs, the system calculates the positive remainder when $T$ is divided by $19$ . At what millisecond mark within the cycle does the event trigger?

Show answer & explanation

Answer: 6

Answer

6

By applying modular arithmetic rules, Fermat's Little Theorem reduces the large exponents. The first term evaluates to

-1 \pmod{19}

and the second term evaluates to

12 \pmod{19}

. Their difference is

-13 \pmod{19}

, which corresponds to a positive remainder of

6

.

Step-by-Step Solution

1

Simplify the first term

37^{45}

modulo 19.

37^{45} \equiv -1 \pmod{19}

Since

37 = 19 \times 2 - 1

, it follows that

37 \equiv -1 \pmod{19}

. An odd power of

-1

is

-1

.

2

Simplify the base of the second term,

40^{23}

, modulo 19.

40^{23} \equiv 2^{23} \pmod{19}

Because

40 = 19 \times 2 + 2

, we can replace the base

40

with its remainder

2

.

3

Use Fermat's Little Theorem to reduce the exponent in

2^{23} \pmod{19}

.

2^{23} \equiv 13 \pmod{19}

Fermat's Little Theorem states

a^{p-1} \equiv 1 \pmod{p}

for a prime

p

. Here,

2^{18} \equiv 1 \pmod{19}

. Therefore,

2^{23} = 2^{18} \times 2^5 \equiv 1 \times 32 \equiv 13 \pmod{19}

.

4

Multiply by 17 and find the remainder of the second term.

17 \times 40^{23} \equiv 12 \pmod{19}

We can write

17 \equiv -2 \pmod{19}

. Then,

(-2) \times 13 = -26

. Adding a multiple of 19 (which is 38) gives

-26 + 38 = 12

.

5

Subtract the second term from the first and convert to a positive remainder.

The final remainder is

6

.

-1 - 12 = -13

. To get the positive remainder, add the modulus 19:

-13 + 19 = 6

.

Key Concept

Applying modular arithmetic rules, properties of negative remainders, and Fermat's Little Theorem to simplify large exponential expressions.

Estimated Time:1m 30s

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