Basic Numeracy Practice Questions — State PSC Exam — Page 13

Question 241Question →

An arithmetic expression is given as $20 \times 26 - 40$ . Which of the following represents the positive remainder when the final computed value of this expression is divided by $7$ ?

0

3

4

6

Show answer & explanation

Answer: 4

Answer

The correct positive remainder is 4.

By following the order of operations, the expression evaluates to 480. When 480 is divided by 7, the quotient is 68, leaving a positive remainder of 4 (

68 \times 7 = 476

, and

480 - 476 = 4

). The same result is efficiently reached using modular arithmetic:

(6 \times 5) - 5 = 25

, and

25 \equiv 4 \pmod 7

.

Step-by-Step Solution

1

Calculate the exact numerical value of the expression.

20 \times 26 - 40 = 520 - 40 = 480

.

Applying standard order of operations (BODMAS), multiplication is performed before subtraction.

2

Divide the computed value by 7 to find the remainder.

480 \div 7 = 68

with a remainder of

4

.

The remainder is the integer left over after dividing the total by the divisor.

3

Verify using modular arithmetic (optional).

20 \equiv -1 \pmod 7

,

26 \equiv 5 \pmod 7

, and

40 \equiv 5 \pmod 7

. Thus,

(-1 \times 5) - 5 = -10

. Modulo 7,

-10 \equiv -3 \equiv 4

.

Modular arithmetic provides a faster secondary method to confirm the remainder without computing large numbers.

Key Concept

Divisibility Rules and Remainder Theorem

Question 242Question →

What is the remainder when the expression $4 + 9 \times 11^{85}$ is divided by $12$ ?

1

5

7

11

Show answer & explanation

Answer: 7

Answer

The correct remainder is 7.

The correct answer is 7 because applying modular arithmetic rules simplifies

11^{85}

to

-1

modulo 12. Substituting this back into the expression yields

4 + 9 \times (-1) = -5

. To find the true positive remainder, the modulus 12 is added to

-5

, resulting in exactly 7.

Step-by-Step Solution

1

Find the remainder of the base 11 when divided by the modulus 12.

11 \equiv -1 \pmod{12}

Using negative remainders simplifies calculations involving large exponents.

2

Evaluate the exponentiation term modulo 12.

11^{85} \equiv (-1)^{85} = -1 \pmod{12}

A negative number raised to an odd power remains negative.

3

Substitute the evaluated power back into the expression following the standard order of operations (BODMAS).

4 + 9 \times (-1) = 4 - 9 = -5 \pmod{12}

Multiplication must be performed before addition.

4

Convert the negative remainder into a valid positive remainder.

-5 + 12 = 7

The final remainder must be a positive integer between 0 and 11, achieved by adding the modulus.

Key Concept

Modular Arithmetic, Negative Remainders, and Order of Operations

Question 243Question →

Consider the basic definitions and categories in the number system. Which of the following statements are mathematically correct? (Select all that apply)

Select all that apply

Every integer can be expressed as a rational number.

If a natural number is not a prime number, it must be a composite number.

The number

2

is the only even prime number.

The fraction

\frac{22}{7}

is exactly equal to the irrational number

\pi

.

Show answer & explanation

Answer: Every integer can be expressed as a rational number.; The number $2$ is the only even prime number.

Answer

The correct statements are that every integer is a rational number, and the number 2 is the only even prime number.

The valid statements accurately reflect mathematical definitions: integers are a subset of rational numbers because any integer can be expressed as a fraction with a denominator of

1

, and

2

is structurally the only even number with exactly two divisors.

Step-by-Step Solution

1

Analyze the relationship between integers and rational numbers.

Every integer

n

can be written as

\frac{n}{1}

.

To verify if an integer fits the definition of a rational number.

2

Evaluate the classification of natural numbers into prime and composite.

The number

1

is an exception, being neither prime nor composite.

To check if the binary classification of non-prime implying composite holds true for all natural numbers.

3

Examine the properties of prime numbers and parity.

The number

2

is prime, while all other even numbers are divisible by

2

and thus composite.

To confirm whether

2

is uniquely the only even prime number.

4

Compare the fraction

\frac{22}{7}

with the mathematical constant

\pi

.

\frac{22}{7}

is a rational approximation, whereas

\pi

is strictly irrational.

To determine if an exact equality exists between these two commonly associated values.

Key Concept

Classification of numbers into rational, irrational, prime, and composite categories, along with their fundamental properties and exceptions.

Question 244Question →

Evaluate the exact positive remainder that results from dividing $5^{75} - 45$ by $126$ .

Show answer & explanation

Answer: 80

Answer

The positive remainder is 80.

The exact remainder is determined using modular arithmetic rules. By noting that

5^3 = 125

, we see that

125 \equiv -1 \pmod{126}

. The expression

5^{75}

can be rewritten as

(5^3)^{25}

, which evaluates to

(-1)^{25} = -1

. Subtracting the constant

45

yields an intermediate remainder of

-46

. Because standard division requires a non-negative remainder, we must add the divisor (

126

) to

-46

, which results in

80

.

Step-by-Step Solution

1

Relate the base of the exponential term to the divisor

5^3 = 125 \equiv -1 \pmod{126}

Finding a power of the base that equals 1 or -1 modulo the divisor significantly simplifies calculations with large exponents.

2

Express the given exponent in terms of the chosen power

5^{75} = (5^3)^{25}

This allows for direct substitution of the modular equivalent.

3

Evaluate the remainder of the exponential portion

(-1)^{25} = -1 \pmod{126}

Substituting

-1

for

125

makes evaluating the

25^{th}

power trivial.

4

Incorporate the constant term into the expression

-1 - 45 = -46 \pmod{126}

The full expression is

5^{75} - 45

, so the constant must be subtracted from the remainder of the exponential part.

5

Convert the negative remainder into a standard positive remainder

-46 + 126 = 80

Remainders must be non-negative. Adding the divisor to a negative remainder provides the correct equivalent positive remainder.

Key Concept

Modular Exponentiation and Negative Remainders

Alternative Method

You could apply the binomial theorem by rewriting

5^{75}

as

125^{25} = (126 - 1)^{25}

. Expanding this yields a series of terms that are all multiples of

126

, plus a final term of

(-1)^{25}

. This clearly leaves a remainder of

-1

. Subtracting

45

gives

-46

, which is then converted to

80

by adding

126

.

Estimated Time:1m 0s

Question 245Question →

The real number system is divided into rational and irrational numbers, with integers forming a subset of rational numbers. Which of the following numbers is classified as a rational number but is NOT an integer?

\frac{15}{4}

-42

\sqrt{25}

\pi

Show answer & explanation

Answer: $\frac{15}{4}$

Answer

\frac{15}{4}

The number

\frac{15}{4}

is a rational number because it is expressed as a ratio of two integers. However, since

15

is not evenly divisible by

4

, it does not simplify to a whole number, meaning it falls strictly within the rational numbers but outside the subset of integers.

Step-by-Step Solution

1

Analyze each number to determine if it is rational or irrational.

\frac{15}{4}

,

-42

, and

\sqrt{25}

can be expressed as ratios of integers, so they are rational.

\pi

is known to be irrational.

The question requires finding a number that is first confirmed to be rational.

2

Identify which of the rational numbers are also integers.

-42

is a negative integer.

\sqrt{25}

simplifies to

5

, which is a positive integer.

Integers include whole numbers and their negatives. We need to eliminate these to find a number that is not an integer.

3

Select the remaining rational number.

\frac{15}{4}

is a rational number that cannot be simplified to a whole number.

It fulfills both conditions: being a rational number and not being an integer.

Key Concept

Classification of rational numbers and integers

Question 246Question →

If $P$ is the smallest prime number and $C$ is the smallest positive composite number, what is the value of $P + C$ ?

Show answer & explanation

Answer: 6

Answer

6

The smallest prime number is 2, as it is the first positive integer greater than 1 with exactly two distinct divisors. The smallest positive composite number is 4, as it is the first positive integer with more than two distinct divisors (1, 2, and 4). The number 1 is excluded from both categories. Therefore, their sum is 2 + 4 = 6.

Step-by-Step Solution

1

Identify the smallest prime number.

P = 2

A prime number is a positive integer greater than 1 that has exactly two positive divisors. The smallest such number is 2.

2

Identify the smallest positive composite number.

C = 4

A composite number is a positive integer that has at least one divisor other than 1 and itself. The numbers 1, 2, and 3 are not composite. The number 4 has divisors 1, 2, and 4, making it the smallest positive composite number.

3

Calculate the sum of P and C.

2 + 4 = 6

The question asks for the sum of the two identified values.

Key Concept

Basic definitions and properties of prime and composite numbers

Question 247Question →

What is the positive remainder when the product $54 \times 82$ is divided by $11$ ?

Show answer & explanation

Answer: 6

Answer

6

By finding the individual remainders of the factors (

-1

and

5

) and multiplying them, we get

-5

. Adding the divisor (

11

) to this negative remainder yields the correct positive remainder of

6

.

Step-by-Step Solution

1

Find the remainder of

54

when divided by

11

.

Since

11 \times 5 = 55

, we can write

54

as

55 - 1

. Thus,

54 \equiv -1 \pmod{11}

.

Using a negative remainder simplifies the subsequent multiplication.

2

Find the remainder of

82

when divided by

11

.

Since

11 \times 7 = 77

, we have

82 - 77 = 5

. Thus,

82 \equiv 5 \pmod{11}

.

We need the remainder of the second factor to apply the remainder theorem for products.

3

Multiply the individual remainders.

(-1) \times 5 = -5

According to modular arithmetic properties, the remainder of a product is the product of the individual remainders.

4

Convert the negative remainder to a positive remainder.

-5 + 11 = 6

The question specifically asks for the positive remainder. Adding the divisor to a negative remainder gives the equivalent positive remainder.

Key Concept

Remainder Theorem and Modular Arithmetic

Question 248Question →

Consider the three fractions $\frac{12}{5}$ , $\frac{24}{7}$ , and $\frac{36}{11}$ . What is the exact value of their Least Common Multiple (LCM)?

Show answer & explanation

Answer: 72

Answer

The Least Common Multiple (LCM) of the given fractions is 72.

To find the Least Common Multiple (LCM) of a set of fractions, we must divide the LCM of their numerators by the HCF of their denominators. The numerators (

12

,

24

, and

36

) have an LCM of

72

. The denominators (

5

,

7

, and

11

) share no common factors other than

1

, making their HCF

1

. Thus, the final LCM is

72 / 1 = 72

.

Step-by-Step Solution

1

State the standard formula for finding the LCM of multiple fractions.

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

This mathematical property governs how multiples apply to rational numbers.

2

Calculate the LCM of the numerators.

The LCM of

12

,

24

, and

36

is

72

.

72

is the smallest integer perfectly divisible by all three numerators.

3

Calculate the HCF of the denominators.

The HCF of

5

,

7

, and

11

is

1

.

The numbers

5

,

7

, and

11

are prime and share no common factors other than

1

.

4

Compute the final fraction LCM.

72 \div 1 = 72

.

Dividing the computed numerator LCM by the denominator HCF yields the answer.

Key Concept

Calculating the LCM of fractions using the specific formula relating numerators and denominators.

Question 249Question →

If the 9-digit number $94x257y36$ is exactly divisible by $88$ , what is the maximum possible value of the mathematical expression $(5x - 3y)$ ?

Show answer & explanation

Answer: 27

Answer

27

By applying the divisibility rules of 8 and 11, we isolate a finite set of possible digit combinations for x and y. Evaluating

(5x - 3y)

across all mathematically valid pairs proves that the pair

x=6, y=1

produces the highest numerical output of 27.

Step-by-Step Solution

1

Determine the co-prime factors of the divisor.

The number must be divisible by 8 and 11, since 88 = 8 * 11 and GCD(8, 11) = 1.

Applying individual divisibility rules for co-prime factors is the standard method for composite divisors.

2

Apply the divisibility rule for 8 to the last three digits (

y36

).

The possible values for

y

are 1, 3, 5, 7, and 9.

A number is divisible by 8 if its last three digits form a multiple of 8. 136, 336, 536, 736, and 936 all yield a remainder of 0 when divided by 8.

3

Apply the divisibility rule for 11 to the entire number.

The expression

(x + y + 4)

must be a multiple of 11.

The rule for 11 requires the difference between the alternating sum of digits to be divisible by 11. Odd positions sum to

(20 + x + y)

and even positions sum to

16

. Difference =

(x + y + 4)

.

4

Find all valid

(x, y)

coordinate pairs.

The valid digit pairs are (6, 1), (4, 3), (2, 5), (0, 7), and (9, 9).

Since

x

and

y

are single digits between 0 and 9,

(x + y + 4)

can only equal 11 or 22. This implies

x + y = 7

or

x + y = 18

.

5

Evaluate the target expression

(5x - 3y)

for all valid pairs.

The values obtained are 27, 11, -5, -21, and 18 respectively. The absolute maximum is 27.

Testing all valid edge cases guarantees that the maximum possible value is correctly identified without assumptions.

Key Concept

Solving for unknown digits using combined co-prime divisibility rules and optimizing a linear algebraic expression based on finite digit constraints.

Estimated Time:2m 30s

Question 250Question →

Determine the exact decimal value of the Least Common Multiple (LCM) for the two fractions $\frac{3}{5}$ and $\frac{9}{10}$ .

Show answer & explanation

Answer: 1.8

Answer

1.8

To find the LCM of fractions, we use the standard formula: LCM(numerators) / HCF(denominators). The numerators are 3 and 9, and their LCM is 9. The denominators are 5 and 10, and their HCF is 5. Dividing the LCM of the numerators by the HCF of the denominators gives 9/5. Converting this fraction to a decimal yields exactly 1.8.

Step-by-Step Solution

1

Identify the correct formula for finding the LCM of fractions.

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

This is the fundamental rule for calculating the least common multiple when dealing with rational numbers.

2

Calculate the Least Common Multiple of the numerators.

The numerators are 3 and 9. Their LCM is 9.

9 is the smallest integer that is perfectly divisible by both 3 and 9.

3

Calculate the Highest Common Factor of the denominators.

The denominators are 5 and 10. Their HCF is 5.

5 is the largest integer that perfectly divides both 5 and 10 without leaving a remainder.

4

Apply the formula and convert the resulting fraction to a decimal.

LCM = 9 / 5 = 1.8

The question specifically asks for the exact decimal value, so the fraction 9/5 must be converted to 1.8.

Key Concept

The Least Common Multiple (LCM) of a set of fractions is found by dividing the LCM of their numerators by the HCF of their denominators.

Question 251Question →

Consider the number $P = 4 + 7 \times 38^{99}$ . What is the remainder when $P$ is divided by $13$ ?

3

11

10

2

Show answer & explanation

Answer: 10

Answer

10

The correct remainder is found by applying modular arithmetic step-by-step. First,

38

modulo

13

simplifies to

-1

. Because

99

is an odd power,

(-1)^{99}

equals

-1

. Multiplying this by

7

yields

-7

. Adding

4

gives an intermediate remainder of

-3

. To find the true positive remainder, we add the divisor

13

to

-3

, resulting in

10

.

Step-by-Step Solution

1

Simplify the base of the exponentiated term modulo 13.

38 \equiv -1 \pmod{13}

Using a negative remainder that is close to zero makes evaluating large exponential powers mathematically manageable.

2

Evaluate the exponentiated term modulo 13.

(-1)^{99} = -1

A negative base raised to an odd power retains its negative sign.

3

Substitute the simplified term back into the original expression and apply the correct order of operations.

4 + 7 \times (-1) = 4 - 7 = -3

Standard order of operations requires multiplication to be performed before addition.

4

Convert the negative intermediate remainder into a valid positive remainder.

-3 \equiv -3 + 13 = 10 \pmod{13}

Remainders must be positive integers strictly less than the divisor.

Key Concept

Modular Arithmetic with Negative Bases and Order of Operations

Alternative Method

One could alternatively find the positive remainder of 38 modulo 13, which is 12, and then evaluate

4 + 7 \times 12^{99} \pmod{13}

. However, this requires observing the cyclicity pattern of powers of 12 modulo 13, making the negative base approach (-1) far more direct and efficient.

Estimated Time:1m 0s

Question 252Question →

The product of two positive integers is $2160$ and their Highest Common Factor (HCF) is $12$ . What is their Least Common Multiple (LCM)?

Show answer & explanation

Answer: 180

Answer

180

For any two positive integers, the product of their Highest Common Factor (HCF) and Least Common Multiple (LCM) is always equal to the product of the numbers themselves. Given that the product is

2160

and the HCF is

12

, the LCM is calculated by rearranging the formula to

\text{LCM} = \frac{\text{Product}}{\text{HCF}}

. Substituting the values gives

2160 \div 12 = 180

.

Step-by-Step Solution

1

Identify the fundamental formula connecting HCF, LCM, and the product of two positive integers.

\text{HCF} \times \text{LCM} = \text{Product}

This relationship allows you to find one missing value when the other two are known.

2

Substitute the known values from the problem into the equation.

12 \times \text{LCM} = 2160

The problem explicitly states that the product is

2160

and the HCF is

12

.

3

Solve for the LCM by dividing both sides of the equation by

12

.

\text{LCM} = 180

Isolating the LCM variable provides the final requested value.

Key Concept

Relationship between the product of two numbers and their HCF and LCM

Question 253Question →

Evaluate the mathematical validity of the three theoretical propositions listed below, which pertain to the fundamental properties of real numbers:

I. For two distinct irrational numbers $x$ and $y$ , it is mathematically possible for both their sum $(x + y)$ and their product $(x \cdot y)$ to evaluate to rational numbers simultaneously.
II. If $N$ represents any natural number, its principal square root $\sqrt{N}$ must be either a natural number or an irrational number; it can never equate to a non-integer rational fraction.
III. The integer $0$ is formally categorized as a positive even number, and the constant $\pi$ is categorized as a rational number because it represents the exact ratio of a circle's circumference to its diameter.

Which of the given propositions is/are correct?

Only I and II

Only I and III

Only II

I, II, and III

Show answer & explanation

Answer: Only I and II

Answer

The correct answer is the option stating that only propositions I and II are correct.

Proposition I is mathematically accurate because using conjugate irrational numbers yields rational sums and products. Proposition II is accurate because the square root of a natural number cannot exist as a non-integer fraction; it must either be an integer or strictly irrational. Proposition III contains two definitive errors: zero is neither positive nor negative, and pi is an irrational number because it cannot be formed by the ratio of two integers.

Step-by-Step Solution

1

Analyze the validity of Proposition I regarding the closure of irrational numbers.

Determined to be True. If

x = 5 + \sqrt{3}

and

y = 5 - \sqrt{3}

(both irrational), their sum is

10

(rational) and their product is

25 - 3 = 22

(rational). Thus, it is mathematically possible.

To test the understanding that the set of irrational numbers is not closed under addition or multiplication.

2

Analyze the validity of Proposition II regarding the square roots of natural numbers.

Determined to be True. A rational fraction in its simplest form

p/q

(where

q > 1

) squared is

p^2/q^2

. This can never simplify to a whole number

N

. Therefore,

\sqrt{N}

is either a whole integer (if

N

is a perfect square) or strictly irrational.

To verify the structural classification of square roots and rational fractions.

3

Analyze the validity of Proposition III regarding the classification of zero and pi.

Determined to be False. First, while zero (

0

) is an even integer, it separates positive and negative numbers and is definitively neither positive nor negative. Second, while

\pi = C/d

, a rational number requires the ratio of two integers. In any true circle, circumference and diameter cannot both be integers simultaneously. Thus,

\pi

is irrational.

To identify specific, common misconceptions regarding zero's sign parity and the rational geometric definition of pi.

Key Concept

Classification and structural properties of real numbers, including the irrationality of pi, the neutrality of zero, and arithmetic operations on irrational numbers.

Estimated Time:2m 0s

Question 254Question →

Two automated watering sprinklers in a greenhouse operate on continuous cycles. Sprinkler X activates every $\frac{12}{5}$ minutes, and Sprinkler Y activates every $\frac{16}{15}$ minutes. If both sprinklers activate simultaneously at a given moment, what is the minimum time interval, in minutes, before they activate together again?

\frac{48}{5}

\frac{4}{15}

\frac{16}{5}

\frac{4}{5}

Show answer & explanation

Answer: $\frac{48}{5}$

Answer

The minimum time interval is

\frac{48}{5}

minutes.

To find when two periodic events will happen together again, we must calculate the Least Common Multiple (LCM) of their periods. For fractions, the LCM is found by dividing the LCM of the numerators (12 and 16, which is 48) by the HCF of the denominators (5 and 15, which is 5). This results in

\frac{48}{5}

.

Step-by-Step Solution

1

Identify that the problem requires finding the Least Common Multiple (LCM) of the two time intervals to determine when the periodic events will align again.

Calculate the LCM of

\frac{12}{5}

and

\frac{16}{15}

.

Simultaneous events occurring at regular intervals coincide at the lowest common multiple of their respective periods.

2

Apply the formula for the LCM of fractions.

Formula:

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

.

This is the standard mathematical rule for finding the LCM of fractional values.

3

Find the LCM of the numerators (12 and 16) and the HCF of the denominators (5 and 15).

LCM(12, 16) = 48. HCF(5, 15) = 5.

The smallest positive number divisible by both 12 and 16 is 48. The largest positive number that divides both 5 and 15 is 5.

4

Substitute the calculated values back into the fraction formula.

\frac{48}{5}

Combining the calculated numerator and denominator yields the final time interval.

Key Concept

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

Question 255Question →

Let $X$ be the total number of positive factors of $120$ , and let $Y$ be the total number of positive factors of $150$ . What is the value of the expression $X + Y \div 2 \times 3$ ?

34

42

18

11

Show answer & explanation

Answer: 34

Answer

The value of the expression is 34.

The correct answer accurately determines the number of factors for 120 (which is 16) and 150 (which is 12). By substituting these values into the expression

16 + 12 \div 2 \times 3

and strictly following the order of operations, division is performed first (

12 \div 2 = 6

), followed by multiplication (

6 \times 3 = 18

), and finally addition (

16 + 18 = 34

).

Step-by-Step Solution

1

Determine the prime factorization of 120 and calculate its total number of positive factors, X.

120 = 2^3 \times 3^1 \times 5^1

. Thus,

X = (3+1)(1+1)(1+1) = 4 \times 2 \times 2 = 16

.

The total number of factors of an integer is found by adding 1 to each exponent in its prime factorization and multiplying the results.

2

Determine the prime factorization of 150 and calculate its total number of positive factors, Y.

150 = 2^1 \times 3^1 \times 5^2

. Thus,

Y = (1+1)(1+1)(2+1) = 2 \times 2 \times 3 = 12

.

Applying the same factor counting formula to the prime factorization of 150.

3

Substitute the values of X and Y into the expression and evaluate according to the order of operations (BODMAS).

Expression:

16 + 12 \div 2 \times 3

. Step 3a (Division):

12 \div 2 = 6

. Step 3b (Multiplication):

6 \times 3 = 18

. Step 3c (Addition):

16 + 18 = 34

.

BODMAS dictates that division and multiplication are evaluated from left to right before any addition is performed.

Key Concept

Calculating total number of factors using prime factorization combined with standard order of operations.

Estimated Time:1m 30s

Question 256Question →

Consider the positive integer $2160$ . How many of its positive factors are perfect squares?

Show answer & explanation

Answer: 6

Answer

6

The prime factorization of

2160

is

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

. For a factor to be a perfect square, all the exponents in its prime factorization must be even integers. Analyzing the bases: the base

2

can have an exponent of

0, 2, \text{ or } 4

(three options). The base

3

can have an exponent of

0 \text{ or } 2

(two options). The base

5

can only have an exponent of

0

to remain even (one option). Multiplying the number of possibilities for each base gives

3 \times 2 \times 1 = 6

total perfect square factors.

Step-by-Step Solution

1

Find the prime factorization of

2160

.

2160 = 2^4 \times 3^3 \times 5^1

Prime factorization is necessary to analyze the properties of the number's factors.

2

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

Any perfect square factor must have the form

2^{2x} \times 3^{2y} \times 5^{2z}

where the exponents are even.

Perfect squares require all prime factors to appear in pairs.

3

Count the number of valid even exponents for each prime factor.

For

2

: exponents

0, 2, 4

(

3

choices). For

3

: exponents

0, 2

(

2

choices). For

5

: exponent

0

(

1

choice).

The exponent of each prime in the factor cannot exceed its exponent in the original number.

4

Calculate the total number of perfect square factors.

3 \times 2 \times 1 = 6

The Fundamental Principle of Counting states we must multiply the independent choices to get the total number of combinations.

Key Concept

Using prime factorization to determine the properties and quantity of specific types of factors (perfect squares).

Question 257Question →

What is the Highest Common Factor (HCF) of the fractions $\frac{3}{4}$ and $\frac{9}{10}$ ?

\frac{3}{20}

\frac{9}{2}

\frac{3}{2}

\frac{9}{20}

Show answer & explanation

Answer: $\frac{3}{20}$

Answer

The correct Highest Common Factor is

\frac{3}{20}

.

To find the Highest Common Factor (HCF) of a set of fractions, the correct formula is to divide the HCF of their numerators by the LCM of their denominators. For the given fractions, the numerators are 3 and 9, and their HCF is 3. The denominators are 4 and 10, and their LCM is 20. Therefore, the HCF of the two fractions is

\frac{3}{20}

.

Step-by-Step Solution

1

Identify the formula for finding the HCF of fractions.

HCF of fractions = (HCF of Numerators) / (LCM of Denominators)

This is the mathematical rule required to find the greatest common divisor for fractional numbers.

2

Calculate the HCF of the numerators (3 and 9).

The HCF of 3 and 9 is 3.

3 is the largest integer that divides both 3 and 9 without leaving a remainder.

3

Calculate the LCM of the denominators (4 and 10).

The LCM of 4 and 10 is 20.

The multiples of 4 are 4, 8, 12, 16, 20... and the multiples of 10 are 10, 20... The lowest common multiple they share is 20.

4

Divide the result of the numerators' HCF by the denominators' LCM.

\frac{3}{20}

Applying the values to the formula gives the final fractional answer.

Key Concept

HCF and LCM of Fractions

Question 258Question →

Three different wooden boards have lengths of $\frac{3}{4}$ meters, $\frac{9}{10}$ meters, and $\frac{15}{8}$ meters, respectively. A carpenter wants to cut them into smaller pieces of equal length such that no material is wasted. What is the maximum possible length of each smaller piece?

\frac{45}{2}

meters

\frac{3}{2}

meters

\frac{3}{40}

meters

\frac{2}{45}

meters

Show answer & explanation

Answer: $\frac{3}{40}$ meters

Answer

The correct answer is

\frac{3}{40}

meters.

To find the maximum equal length that can be cut from all three boards, we must find the Highest Common Factor (HCF) of their lengths. The formula for the HCF of fractions is the HCF of the numerators divided by the LCM of the denominators. The numerators are

3, 9,

and

15

, yielding an HCF of

3

. The denominators are

4, 10,

and

8

, yielding an LCM of

40

. Thus, the maximum length is

\frac{3}{40}

meters.

Step-by-Step Solution

1

Identify the mathematical operation required.

The problem asks for the 'maximum possible length' that can evenly divide all three board lengths without remainders. This requires finding the Highest Common Factor (HCF) of the three fractions.

Cutting multiple items into the largest possible equal smaller pieces is a classic application of HCF.

2

Recall the formula for finding the HCF of fractions.

\text{HCF of fractions} = \frac{\text{HCF of Numerators}}{\text{LCM of Denominators}}

This is the standard rule for finding the greatest common divisor of a set of rational numbers.

3

Calculate the HCF of the numerators (

3, 9, 15

).

The factors of

3

are

1, 3

. The factors of

9

are

1, 3, 9

. The factors of

15

are

1, 3, 5, 15

. The highest common factor is

3

.

The numerator of our final fraction must be the HCF of the given numerators.

4

Calculate the LCM of the denominators (

4, 10, 8

).

The prime factorization of

4

is

2^2

. For

8

, it is

2^3

. For

10

, it is

2 \times 5

. The LCM takes the highest power of each prime present:

2^3 \times 5 = 8 \times 5 = 40

.

The denominator of our final fraction must be the LCM of the given denominators.

5

Combine the results to find the final HCF.

\frac{3}{40}

meters.

Applying the formula from Step 2 with the values found in Step 3 and Step 4.

Key Concept

HCF and LCM of Fractions

Question 259Question →

Consider the following $8$ mathematical expressions:

I. $\pi - \frac{22}{7}$
II. $\frac{\sqrt{27}}{\sqrt{3}}$
III. $(\sqrt{3} - \sqrt{2})^2$
IV. The infinite decimal $0.101101110\dots$ (where the number of consecutive $1$ s increases by one each time)
V. $1.414$
VI. $\sqrt{10} \times \sqrt{2.5}$
VII. $e^0$
VIII. $\sqrt{2} + \sqrt{8}$

How many of the above expressions evaluate to a rational number?

Show answer & explanation

Answer: 4

Answer

Exactly 4 of the expressions evaluate to a rational number.

Out of the 8 expressions provided, exactly 4 result in rational numbers: Expression II (simplifies to 3), Expression V (terminating decimal 1414/1000), Expression VI (simplifies to sqrt(25) = 5), and Expression VII (simplifies to 1). The remaining 4 expressions yield irrational results.

Step-by-Step Solution

1

Evaluate expression I (pi minus 22/7)

The expression evaluates to an irrational number.

The number pi is irrational, while 22/7 is a rational approximation. The difference between an irrational number and a rational number is always irrational.

2

Evaluate expression II (sqrt(27) / sqrt(3))

The expression simplifies to 3, which is a rational number.

Using the properties of radicals, sqrt(27) / sqrt(3) = sqrt(27/3) = sqrt(9) = 3.

3

Evaluate expression III (square of (sqrt(3) - sqrt(2)))

The expression expands to 5 - 2*sqrt(6), which is an irrational number.

Applying the binomial square formula (a-b)^2 = a^2 - 2ab + b^2 yields 3 - 2*sqrt(6) + 2. Since sqrt(6) is irrational, the entire expression is irrational.

4

Evaluate expression IV (the pattern decimal 0.101101110...)

The expression is an irrational number.

The decimal is non-terminating and non-periodic (the pattern changes constantly by adding an extra 1), which is the definition of an irrational decimal.

5

Evaluate expression V (1.414)

The expression is a rational number.

1.414 is a terminating decimal. Any terminating decimal can be written as a fraction of integers (1414/1000), making it rational.

6

Evaluate expression VI (sqrt(10) * sqrt(2.5))

The expression simplifies to 5, which is a rational number.

Multiplying the terms inside the radicals gives sqrt(10 * 2.5) = sqrt(25) = 5.

7

Evaluate expression VII (e^0)

The expression simplifies to 1, which is a rational number.

Any non-zero real number raised to the power of 0 equals 1.

8

Evaluate expression VIII (sqrt(2) + sqrt(8))

The expression simplifies to 3*sqrt(2), which is an irrational number.

sqrt(8) can be simplified to 2*sqrt(2). Adding sqrt(2) gives 3*sqrt(2), which remains an irrational product.

9

Count the total number of rational expressions

Expressions II, V, VI, and VII are rational. Total count is 4.

Identifying the rational outcomes from the previous steps.

Key Concept

Properties and Definitions of Rational and Irrational Numbers

Estimated Time:2m 30s

Question 260Question →

Analyze the following three mathematical assertions concerning the classification and properties of numbers:

Assertion I: The expression $\sqrt{32} - 16 \div 4 \times \sqrt{2}$ evaluates to an irrational number.
Assertion II: For any prime number $p$ that is strictly greater than $2$ , the value of $p^2 - 1$ is always a multiple of $8$ .
Assertion III: The number $0$ is classified as a rational number and an even integer, but it is neither a prime nor a composite number.

Which of the assertions provided above are logically correct?

Assertions II and III are the only correct statements.

Assertions I, II, and III are all correct statements.

Assertion III is the only correct statement.

Assertions I and II are the only correct statements.

Show answer & explanation

Answer: Assertions II and III are the only correct statements.

Answer

Assertions II and III are the only correct statements.

Assertion II is mathematically accurate because the square of any odd prime minus one always factors into

4k(k+1)

, which is a multiple of

8

. Assertion III is also accurate as zero strictly fits the definitions of a rational number (

0/1

) and an even integer (

0 = 2 \times 0

), while being neither prime nor composite. Assertion I is false because correctly evaluating the expression from left-to-right yields

0

, which is a rational number.

Step-by-Step Solution

1

Evaluate Assertion I using the correct order of operations (BODMAS/PEMDAS).

\sqrt{32} - 16 \div 4 \times \sqrt{2} = 4\sqrt{2} - 4 \times \sqrt{2} = 4\sqrt{2} - 4\sqrt{2} = 0

.

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

0

, is a rational number, making Assertion I false.

2

Analyze Assertion II for primes

p > 2

.

All primes

p > 2

are odd, so

p = 2k + 1

for some integer

k

. Then

p^2 - 1 = (2k+1)^2 - 1 = 4k^2 + 4k = 4k(k+1)

.

Since

k

and

k+1

are consecutive integers, one must be even. Thus,

k(k+1)

is an even integer, making

4k(k+1)

a multiple of

8

. Assertion II is strictly true.

3

Verify the classifications of the number

0

in Assertion III.

0

can be written as

0/1

(rational), is divisible by

2

without remainder (even), and has no strictly positive divisors that satisfy the definition of primes or composites.

By standard mathematical definitions,

0

is rational, even, and neither prime nor composite. Assertion III is true.

Key Concept

Classification of numbers involving rational/irrational properties, algebraic properties of prime numbers, and the precise classification of zero.

Alternative Method

For Assertion II, students under time pressure can test the first few qualifying prime numbers (e.g.,

p=3 \Rightarrow 3^2-1=8

;

p=5 \Rightarrow 5^2-1=24

) to quickly establish confidence in the truth of the statement without a formal algebraic proof.

Estimated Time:2m 0s

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