All practice questions

What is the unit digit of the expression $K = (233^{85} \times 437^{94}) - 168^{47}$ ?

Show answer & explanation

Answer: 5

Answer

The unit digit of the given expression is 5.

To find the unit digit of

K = (233^{85} \times 437^{94}) - 168^{47}

, analyze each term using base cyclicity rules:
1.

233^{85}

: Base digit 3 has cyclicity 4.

85 \div 4 = 21

remainder 1.

3^1 = 3

.
2.

437^{94}

: Base digit 7 has cyclicity 4.

94 \div 4 = 23

remainder 2.

7^2 = 49 \rightarrow 9

.
3. Product unit digit:

3 \times 9 = 27 \rightarrow 7

.
4.

168^{47}

: Base digit 8 has cyclicity 4.

47 \div 4 = 11

remainder 3.

8^3 = 512 \rightarrow 2

.
5. Subtracting gives

7 - 2 = 5

.

Step-by-Step Solution

1

Find the unit digit of

233^{85}

Unit digit is 3

The unit digit of powers of 3 follows a cycle of 4: (3, 9, 7, 1). Since

85 \equiv 1 \pmod 4

, the unit digit is

3^1 = 3

.

2

Find the unit digit of

437^{94}

Unit digit is 9

The unit digit of powers of 7 follows a cycle of 4: (7, 9, 3, 1). Since

94 \equiv 2 \pmod 4

, the unit digit is

7^2 = 49 \implies 9

.

3

Find the unit digit of the product

(233^{85} \times 437^{94})

Unit digit is 7

Multiplying the unit digits of the two terms gives

3 \times 9 = 27

, which has a unit digit of 7.

4

Find the unit digit of

168^{47}

Unit digit is 2

The unit digit of powers of 8 follows a cycle of 4: (8, 4, 2, 6). Since

47 \equiv 3 \pmod 4

, the unit digit is

8^3 = 512 \implies 2

.

5

Calculate the final unit digit of the composite expression

5

Subtracting the unit digit of the subtracted term from the product's unit digit yields

7 - 2 = 5

.

Key Concept

Unit Digit and Cyclicity

Estimated Time:1m 30s

Question 802Question →

Determine the unit digit of the finite series and exponent tower expression $S = \sum_{k=1}^{50} (k!)^{k!} + 7^{7^7}$ .

Show answer & explanation

Answer: 0

Answer

The unit digit of the expression is 0.

The sum of the factorial terms from k=1 to k=4 contributes unit digits of 1, 4, 6, and 6, totaling 17 (unit digit 7). Terms for k >= 5 end in 0. The power tower term 7^(7^7) has an exponent 7^7 which leaves a remainder of 3 when divided by 4. Since the cyclicity of 7 is 4, 7^3 gives a unit digit of 3. Adding 7 and 3 results in 10, giving a final unit digit of 0.

Step-by-Step Solution

1

Calculate unit digits of factorial terms for k from 1 to 4

Unit digits are 1, 4, 6, and 6 respectively

1! = 1, 2! = 2, 3! = 6, 4! = 24 (with base ending in 4 raised to an even power 24 giving unit digit 6)

2

Analyze factorial terms for k >= 5

Unit digit is 0 for all k >= 5

Factorials for k >= 5 contain factors 2 and 5, making the trailing digit 0

3

Sum the unit digits of the series sum

Sum of unit digits is 1 + 4 + 6 + 6 = 17, giving unit digit 7

Only terms from k = 1 to 4 contribute to the unit digit of the factorial sum

4

Find the unit digit of the power tower 7^(7^7) using cyclicity modulo 4

7^7 mod 4 = 3, so unit digit is 7^3 mod 10 = 3

The base 7 has a cyclicity of 4, and 7^7 is congruent to 3 modulo 4

5

Add the unit digits of both components

7 + 3 = 10, unit digit is 0

Combining the unit digit of the factorial sum (7) and the tower exponent term (3)

Key Concept

Combining factorial unit digit termination properties with exponent tower cyclicity modulo 4.

Question 803Question →

What value is obtained upon simplifying the mathematical expression given below using the correct order of operations (BODMAS)?

87.5\% \text{ of } 120 - \left[ 2 \frac{2}{5} \times \left\{ 24.5 - \left( 4.5 \div \overline{2.4 - 0.9} \times 1.6 + 1 \frac{1}{5} \right) \right\} \right]

60.6

66.36

53.58

52.2

Show answer & explanation

Answer: $60.6$

Answer

60.6

The correct result

60.6

is achieved by systematically resolving operations from the innermost group outward: first the bar/vinculum, followed by division and multiplication inside the parentheses, subtraction in the curly brackets, multiplication in the square brackets, and finally subtracting this total from

87.5\%

of

120

.

Step-by-Step Solution

1

Evaluate the expression under the vinculum (bar)

\overline{2.4 - 0.9} = 1.5

The vinculum has the highest priority among grouping symbols.

2

Evaluate the terms inside the parentheses

(...)

using BODMAS

4.5 \div 1.5 \times 1.6 + 1 \frac{1}{5} = 3 \times 1.6 + 1.2 = 4.8 + 1.2 = 6.0

Perform division first (

4.5 \div 1.5 = 3

), then multiplication (

3 \times 1.6 = 4.8

), and finally addition (

4.8 + 1.2 = 6.0

).

3

Simplify the expression inside the curly brackets

\{...\}

24.5 - 6.0 = 18.5

Subtract the result of the parentheses from

24.5

.

4

Simplify the expression inside the square brackets

[...]

2 \frac{2}{5} \times 18.5 = 2.4 \times 18.5 = 44.4

Convert the mixed fraction

2 \frac{2}{5}

to decimal

2.4

and multiply by

18.5

.

5

Calculate the percentage value

87.5\% \text{ of } 120 = \frac{7}{8} \times 120 = 105

Convert

87.5\%

to the equivalent fraction

\frac{7}{8}

and multiply by

120

.

6

Perform the final subtraction

105 - 44.4 = 60.6

Subtract the value of the square brackets from the percentage result.

Key Concept

Strict evaluation of nested brackets (vinculum, round, curly, square) and mixed operators (percentage, fraction, division, multiplication) using the BODMAS rule.

Question 804Question →

What is the final numerical value of the following expression evaluated using the standard BODMAS rule?

50 - [18 \div 3 \times (4 + 2)]

Show answer & explanation

Answer: 14

Answer

The final simplified value of the expression is 14.

Following the BODMAS order of operations: first simplify the inner bracket

(4 + 2) = 6

. Next, evaluate inside the square bracket from left to right:

18 \div 3 = 6

, and then

6 \times 6 = 36

. Finally, subtract from 50 to get

50 - 36 = 14

.

Step-by-Step Solution

1

Evaluate the innermost round brackets

4 + 2 = 6

According to BODMAS, operations inside brackets must be performed first.

2

Perform division inside the square brackets

18 \div 3 = 6

Division and multiplication have equal priority and are performed from left to right.

3

Perform multiplication inside the square brackets

6 \times 6 = 36

Multiply the quotient obtained from division by the bracketed sum.

4

Subtract the result inside brackets from 50

50 - 36 = 14

Perform subtraction as the final operation.

Key Concept

BODMAS / Order of Operations

Question 805Question →

What is the unit digit of the composite expression $E = (1! + 2! + 3! + \dots + 20!)^{2026} + (3^1 \times 3^2 \times 3^3 \times \dots \times 3^{20})$ ?

Show answer & explanation

Answer: 8

Answer

The unit digit of the expression is 8.

Evaluating the expression requires breaking it down into two components. First, for the factorial sum

1! + 2! + 3! + \dots + 20!

, every term from

5!

onward contains factors of both 2 and 5, so its unit digit is 0. The unit digit of the sum is determined solely by

1! + 2! + 3! + 4! = 33

, which has a unit digit of 3. Raising 3 to the power 2026 gives

3^{2026}

. Since the unit digits of powers of 3 repeat in cycles of 4 (3, 9, 7, 1) and

2026 \equiv 2 \pmod 4

, the unit digit of

3^{2026}

is

3^2 = 9

.

Second, the product

3^1 \times 3^2 \times \dots \times 3^{20}

simplifies using the exponent addition rule to

3^{1+2+\dots+20} = 3^{210}

. Dividing 210 by 4 leaves a remainder of 2, so

3^{210}

also has a unit digit of

3^2 = 9

.

Adding the unit digits of both terms gives

9 + 9 = 18

, resulting in a final unit digit of 8.

Step-by-Step Solution

1

Find the unit digit of the inner factorial sum

S = 1! + 2! + 3! + \dots + 20!

.

The unit digit of

S

is 3.

For all

k \ge 5

,

k!

is divisible by 10 and ends in 0. Thus, only the sum of the first four terms

1! + 2! + 3! + 4! = 1 + 2 + 6 + 24 = 33

determines the unit digit.

2

Calculate the unit digit of the first term

S^{2026} \equiv 3^{2026}

.

The unit digit of the first term is 9.

The unit digits of powers of 3 repeat in a cycle of 4 (3, 9, 7, 1). Dividing the exponent 2026 by 4 gives a remainder of 2 (

2026 = 4 \times 506 + 2

). Therefore, the unit digit is

3^2 = 9

.

3

Simplify the exponential product

P = 3^1 \times 3^2 \times 3^3 \times \dots \times 3^{20}

.

The product simplifies to

3^{210}

.

By exponent multiplication rules,

3^1 \times 3^2 \times \dots \times 3^{20} = 3^{\sum_{i=1}^{20} i}

. The sum of the first 20 positive integers is

\frac{20 \times 21}{2} = 210

.

4

Calculate the unit digit of

3^{210}

.

The unit digit of the second term is 9.

Dividing the exponent 210 by 4 gives a remainder of 2 (

210 = 4 \times 52 + 2

). Therefore, the unit digit is

3^2 = 9

.

5

Combine the unit digits of the two terms.

The unit digit of

E

is 8.

Adding the unit digits gives

9 + 9 = 18

. The unit digit of 18 is 8.

Key Concept

Unit Digit and Cyclicity of Factorial and Exponential Expressions

Estimated Time:2m 0s

Question 806Question →

What exact numerical result is obtained when simplifying the following multi-bracket expression according to the BODMAS order of operations?

75\% \text{ of } 240 - \left[ 38.5 + \left\{ 15\frac{1}{2} - \left( 6.8 \times 2.5 - \overline{4.2 + 3.8} \right) \div 0.5 \right\} \right]

Show answer & explanation

Answer: 144

Answer

The final simplified numerical value of the given expression is 144.

Following the BODMAS rule strictly: first evaluate the vinculum

\overline{4.2 + 3.8} = 8

; then resolve round brackets

(6.8 \times 2.5 - 8 = 9)

; next resolve curly brackets

(15.5 - 9 \div 0.5 = -2.5)

; then square brackets

(38.5 - 2.5 = 36)

; and finally compute

75\% \text{ of } 240 = 180

. Subtracting

36

from

180

gives the correct result of 144.

Step-by-Step Solution

1

Evaluate the vinculum (bar) operator first

\overline{4.2 + 3.8} = 8

According to BODMAS, the vinculum takes precedence over round brackets.

2

Simplify the terms inside the round brackets (parentheses)

6.8 \times 2.5 - 8 = 17 - 8 = 9

Multiplication precedes subtraction inside the innermost round brackets.

3

Simplify the terms inside the curly brackets (braces)

15.5 - (9 \div 0.5) = 15.5 - 18 = -2.5

Division (

9 \div 0.5 = 18

) is performed before subtracting from

15.5

(

15\frac{1}{2}

).

4

Simplify the terms inside the square brackets

38.5 + (-2.5) = 36

Adding the negative result from the curly brackets to

38.5

yields

36

.

5

Calculate the percentage 'Of' operation and perform final subtraction

75\% \text{ of } 240 - 36 = 180 - 36 = 144

'Of' operation (

75\% \times 240 = 180

) is calculated before subtracting the bracketed total.

Key Concept

BODMAS Rule with Vinculum and Nested Brackets

Estimated Time:2m 0s

Question 807Question →

What is the result when the mathematical expression

45 - [12 + (8 \div 2 \times 3)]

is simplified strictly adhering to the standard BODMAS order of operations?

Show answer & explanation

Answer: 21

Answer

21

Following the BODMAS rule, operations inside the innermost parentheses are solved first from left to right:

8 \div 2 = 4

, then

4 \times 3 = 12

. Next, addition inside the square brackets gives

12 + 12 = 24

. Finally, subtracting 24 from 45 yields 21.

Step-by-Step Solution

1

Evaluate division inside the parentheses

8 ÷ 2 = 4

Division and multiplication have equal priority and are evaluated from left to right inside parentheses.

2

Evaluate multiplication inside the parentheses

4 × 3 = 12

Completing the operations within the parentheses.

3

Add the values inside the square brackets

12 + 12 = 24

Evaluating the bracketed addition.

4

Subtract from the outer term

45 - 24 = 21

Performing final subtraction outside the brackets.

Key Concept

BODMAS Rule (Order of Operations)

Estimated Time:45s

Question 808Question →

What is the unit digit of the composite exponential expression $E = 433^{45} + 818^{80} - 625^{75}$ ?

4

9

2

6

Show answer & explanation

Answer: 4

Answer

4

Evaluating each term individually:

433^{45}

has unit digit

3

(since

45 \pmod 4 = 1

),

818^{80}

has unit digit

6

(since

80 \pmod 4 = 0

, giving the 4th power unit digit

8^4 \rightarrow 6

), and

625^{75}

has unit digit

5

. Combining these gives

3 + 6 - 5 = 4

.

Step-by-Step Solution

1

Find the unit digit of

433^{45}

Unit digit is 3

The unit digit of base 433 is 3. The cyclicity of 3 is 4 (

3^1=3, 3^2=9, 3^3=7, 3^4=1

). Dividing the exponent 45 by 4 gives

45 = 4 \times 11 + 1

(remainder 1). Thus, the unit digit is

3^1 = 3

.

2

Find the unit digit of

818^{80}

Unit digit is 6

The unit digit of base 818 is 8. The cyclicity of 8 is 4 (

8^1=8, 8^2=4, 8^3=2, 8^4=6

). Dividing exponent 80 by 4 gives a remainder of 0. When remainder is 0, we take the 4th power, giving unit digit

6

(from

8^4 = 4096

).

3

Find the unit digit of

625^{75}

Unit digit is 5

The unit digit of base 625 is 5. Any positive integer power of a number ending in 5 always ends in 5 (cyclicity of 1).

4

Combine the unit digits of all terms

4

Substitute the individual unit digits into the expression:

3 + 6 - 5 = 4

.

Key Concept

Unit Digit and Cyclicity Rules for Exponential Expressions

Question 809Question →

Is the positive integer $n$ divisible by $36$ ?

Statement I: $n^2$ is divisible by $108$ .
Statement II: $n^3$ is divisible by $576$ .

Which of the following options correctly describes the sufficiency of the statements?

Statement I alone is sufficient to answer the question, but Statement II alone is not sufficient.

Statement II alone is sufficient to answer the question, but Statement I alone is not sufficient.

Either Statement I alone or Statement II alone is sufficient to answer the question.

Neither Statement I nor Statement II is sufficient to answer the question, and additional data is needed.

Both Statement I and Statement II together are sufficient to answer the question, but neither statement alone is sufficient.

Show answer & explanation

Answer: Both Statement I and Statement II together are sufficient to answer the question, but neither statement alone is sufficient.

Answer

Both Statement I and Statement II together are sufficient to answer the question, but neither statement alone is sufficient.

Evaluating both statements together shows that Statement I requires

n

to be a multiple of

18

(

2^1 \times 3^2

) and Statement II requires

n

to be a multiple of

12

(

2^2 \times 3^1

). The least common multiple of

18

and

12

is

36

, which proves that

n

is guaranteed to be divisible by

36

. Neither statement alone provides both prime factor requirements.

Step-by-Step Solution

1

Analyze Statement I individually

Statement I states that

n^2

is divisible by

108 = 2^2 \times 3^3

. In the prime factorization of a square

n^2

, all exponents must be even numbers. Thus,

n^2

must contain at least

2^2

and

3^4

. Taking square roots,

n

must be a multiple of

2^1 \times 3^2 = 18

. If

n = 18

,

n

is NOT divisible by

36

. If

n = 36

,

n

IS divisible by

36

. Because we get both 'No' and 'Yes' answers, Statement I alone is NOT sufficient.

We must test if Statement I uniquely determines whether

n

is divisible by

36

.

2

Analyze Statement II individually

Statement II states that

n^3

is divisible by

576 = 2^6 \times 3^2

. In the prime factorization of a cube

n^3

, all exponents must be multiples of

3

. Thus,

n^3

must contain at least

2^6

and

3^3

. Taking cube roots,

n

must be a multiple of

2^2 \times 3^1 = 12

. If

n = 12

,

n

is NOT divisible by

36

. If

n = 36

,

n

IS divisible by

36

. Because we get both 'No' and 'Yes' answers, Statement II alone is NOT sufficient.

We must test if Statement II uniquely determines whether

n

is divisible by

36

.

3

Combine Statement I and Statement II

From Statement I,

n

contains at least

3^2

in its prime factorization. From Statement II,

n

contains at least

2^2

in its prime factorization. Combining these requirements,

n

must contain at least

2^2 \times 3^2 = 36

. Therefore,

n

is guaranteed to be divisible by

36

. Both statements together yield a definitive 'Yes'.

Evaluating both statements together combines the minimal necessary powers of each prime factor.

Key Concept

Data Sufficiency evaluation of prime factor exponents and divisibility rules

Estimated Time:2m 0s

Question 810Question →

Determine the unit digit of the numerical expression $N = (333^{444} \times 444^{333}) + (777^{888} \times 888^{777}) - 222^{555}$ .

Show answer & explanation

Answer: 4

Answer

The unit digit of the given expression is 4.

By analyzing cyclicity of each base (

3, 4, 7, 8, 2

), we find that

333^{444} \rightarrow 1

,

444^{333} \rightarrow 4

,

777^{888} \rightarrow 1

,

888^{777} \rightarrow 8

, and

222^{555} \rightarrow 8

. Thus, the overall unit digit is

(1 \times 4) + (1 \times 8) - 8 = 4 + 8 - 8 = 4

.

Step-by-Step Solution

1

Calculate unit digit of

333^{444} \times 444^{333}

Unit digit is 4

Base 3 has cyclicity 4;

444 \pmod 4 = 0 \implies 3^4 \rightarrow 1

. Base 4 has cyclicity 2; odd exponent

333 \implies 4^1 \rightarrow 4

. Total unit digit for term 1 =

1 \times 4 = 4

.

2

Calculate unit digit of

777^{888} \times 888^{777}

Unit digit is 8

Base 7 has cyclicity 4;

888 \pmod 4 = 0 \implies 7^4 \rightarrow 1

. Base 8 has cyclicity 4;

777 \pmod 4 = 1 \implies 8^1 \rightarrow 8

. Total unit digit for term 2 =

1 \times 8 = 8

.

3

Calculate unit digit of

222^{555}

Unit digit is 8

Base 2 has cyclicity 4;

555 \pmod 4 = 3 \implies 2^3 \rightarrow 8

.

4

Evaluate combined expression unit digit

4

Combine term unit digits:

(4 + 8 - 8) = 4

.

Key Concept

Unit Digit and Cyclicity of Numbers

Question 811Question →

What is the simplified value of the following mathematical expression evaluated using the standard BODMAS rule?

24 + 16 \div 4 \times 2 - (7 - \overline{5 - 2})

28

22

16

32

Show answer & explanation

Answer: 28

Answer

28

Following the standard order of operations (BODMAS), we first simplify the expression under the vinculum bar:

5 - 2 = 3

. Next, we complete the subtraction inside the parentheses:

7 - 3 = 4

. Then, we resolve division and multiplication from left to right:

16 \div 4 = 4

, and

4 \times 2 = 8

. Finally, combining addition and subtraction gives

24 + 8 - 4 = 28

.

Step-by-Step Solution

1

Evaluate the expression under the vinculum (bar)

\overline{5 - 2} = 3

The vinculum takes the highest priority in bracket operations.

2

Evaluate the round brackets

(7 - 3) = 4

Operations within brackets must be solved next.

3

Perform Division and Multiplication from left to right

16 \div 4 = 4; \quad 4 \times 2 = 8

Division and multiplication share equal precedence and are performed in order from left to right.

4

Perform Addition and Subtraction from left to right

24 + 8 - 4 = 28

Final addition and subtraction operations are executed left-to-right.

Key Concept

BODMAS Rule with Vinculum (Order of Operations)

Question 812Question →

What is the unit digit of the numerical expression $N = (137^{137} \times 264^{102}) + 619^{199}$ ?

Show answer & explanation

Answer: 1

Answer

1

The unit digit of

137^{137}

is derived from

137 \pmod 4 = 1

, giving

7^1 = 7

. The unit digit of

264^{102}

is 6 because the exponent 102 is even. Multiplying these unit digits gives

7 \times 6 = 42

, which contributes a unit digit of 2. The unit digit of

619^{199}

is 9 because 199 is odd. Adding the unit digits yields

2 + 9 = 11

, whose unit digit is 1.

Step-by-Step Solution

1

Determine the unit digit of

137^{137}

using the cyclicity of 7.

The exponent 137 leaves a remainder of 1 when divided by 4, giving a unit digit of

7^1 = 7

.

The unit digit pattern for powers of 7 repeats every 4 powers (7, 9, 3, 1).

2

Determine the unit digit of

264^{102}

using the cyclicity of 4.

Since 102 is an even exponent,

4^{\text{even}}

yields a unit digit of 6.

Powers of 4 end in 4 for odd exponents and 6 for even exponents.

3

Calculate the unit digit of the product

(137^{137} \times 264^{102})

.

(7 \times 6) = 42

, which has a unit digit of 2.

The unit digit of a product depends only on the product of the unit digits of its factors.

4

Determine the unit digit of

619^{199}

using the cyclicity of 9.

Since 199 is an odd exponent,

9^{\text{odd}}

yields a unit digit of 9.

Powers of 9 end in 9 for odd exponents and 1 for even exponents.

5

Sum the unit digits to find the final unit digit of expression

N

.

2 + 9 = 11

, which has a unit digit of 1.

The unit digit of the sum of two terms is the unit digit of the sum of their individual unit digits.

Key Concept

Unit digit determination using exponent cyclicity rules for base digits 4, 7, and 9.

Estimated Time:1m 15s

Question 813Question →

If $M = \sqrt{7 + 2\sqrt{10}} + \sqrt{7 - 2\sqrt{10}}$ and $x$ satisfies the exponential equation $\left(\sqrt[3]{4}\right)^{2x-1} = \left(\frac{1}{16}\right)^{x-4}$ with $\left(\sqrt[3]{4}\right)^{2x-1} = 2^k$ , what is the value of $M^2 + k$ ?

Show answer & explanation

Answer: 23.5

Answer

The value of

M^2 + k

is 23.5.

By writing

7 \pm 2\sqrt{10}

as

(\sqrt{5} \pm \sqrt{2})^2

, the radical simplifies cleanly to

M = 2\sqrt{5}

, giving

M^2 = 20

. Rewriting the index equation in terms of base 2 yields

\frac{4x-2}{3} = -4x + 16

, which gives

x = \frac{25}{8}

and exponent

k = 3.5

. Adding

M^2

and

k

results in 23.5.

Step-by-Step Solution

1

Simplify the nested surd expression for M

M = 2\sqrt{5}, so M^2 = 20

Recognize that

7 \pm 2\sqrt{10} = (\sqrt{5} \pm \sqrt{2})^2

.

2

Convert both sides of the exponential equation to base 2

2^{\frac{4x-2}{3}} = 2^{-4x+16}

Apply laws of indices:

\sqrt[3]{4} = 2^{2/3}

and

\frac{1}{16} = 2^{-4}

.

3

Solve for x by equating the powers of 2

x = \frac{25}{8} = 3.125

Since bases are equal, the powers must be equal.

4

Determine the exponent value k

k = 3.5

Substitute x into the exponent expression

k = -4x + 16

.

5

Calculate the final combined expression M^2 + k

23.5

Add

M^2 = 20

and

k = 3.5

.

Key Concept

Nested radical simplification using binomial square expansion combined with solving exponential equations via prime base unification.

Question 814Question →

The Highest Common Factor (HCF) and Least Common Multiple (LCM) of two positive three-digit integers $P$ and $Q$ (where $P > Q$ ) are $24$ and $1080$ , respectively. If the difference between the two numbers is $96$ , what is the sum of the two numbers ( $P + Q$ )?

Show answer & explanation

Answer: 336

Answer

The sum of the two numbers P and Q is 336.

By representing

P = 24a

and

Q = 24b

with

\gcd(a, b) = 1

, the relation

\text{HCF} \times \text{LCM} = P \times Q

yields

a \times b = 45

. The coprime factor pairs of

45

are

(45, 1)

and

(9, 5)

. The pair

(45, 1)

gives

1080

and

24

, which are not both three-digit numbers. The pair

(9, 5)

gives

P = 216

and

Q = 120

, both of which are three-digit numbers with a difference of

96

. The sum of these two numbers is

216 + 120 = 336

.

Step-by-Step Solution

1

Express the two numbers in terms of their HCF and coprime factors

Let

P = 24a

and

Q = 24b

where

\gcd(a, b) = 1

and

a > b

.

Any two numbers sharing an HCF of

h

can be represented as

h \cdot a

and

h \cdot b

where

a

and

b

have no common prime factors.

2

Relate the product of the coprime factors to the LCM and HCF

a \times b = \frac{\text{LCM}}{\text{HCF}} = \frac{1080}{24} = 45

.

The product of two numbers equals the product of their HCF and LCM:

(24a)(24b) = 24 \times 1080

.

3

Identify all coprime factor pairs of 45

The coprime factor pairs

(a, b)

with

a > b

are

(45, 1)

and

(9, 5)

.

Factor pairs such as

(15, 3)

are invalid because

\gcd(15, 3) = 3 \neq 1

.

4

Apply the three-digit integer and difference constraints to select the valid pair

For

(a, b) = (9, 5)

,

P = 24 \times 9 = 216

and

Q = 24 \times 5 = 120

. Difference =

216 - 120 = 96

.

The pair

(45, 1)

yields

P = 1080

(four digits) and

Q = 24

(two digits), failing the three-digit criteria.

5

Calculate the sum of

P

and

Q

P + Q = 216 + 120 = 336

.

The question asks specifically for the sum

P + Q

.

Key Concept

Properties of HCF and LCM including HCF x LCM = Product of Numbers and Prime Factor Coprimality

Question 815Question →

What is the simplified numerical value of the following mathematical expression when evaluated strictly according to the BODMAS rule?

45 - \left[ 3 \times \left\{ 12 \div \left( 1.5 + \overline{2.7 - 1.2} \right) \right\} + 2.5 \right] \times 2

16

6

61

-17

Show answer & explanation

Answer: 16

Answer

16

Evaluating the expression according to VBODMAS rules step-by-step: first the vinculum gives 1.5; adding inside round brackets gives 3; dividing inside curly braces gives 4; multiplying and adding inside square brackets gives 14.5; finally multiplying by 2 gives 29, and subtracting from 45 yields 16.

Step-by-Step Solution

1

Evaluate the expression under the vinculum (bar)

\overline{2.7 - 1.2} = 1.5

According to VBODMAS, operations under a vinculum must be simplified first.

2

Simplify the expression inside the round brackets (parentheses)

1.5 + 1.5 = 3

Parentheses have priority after the vinculum is resolved.

3

Evaluate the division inside the curly braces

12 \div 3 = 4

Operations within braces are evaluated next.

4

Simplify the expression inside the square brackets

3 \times 4 + 2.5 = 12 + 2.5 = 14.5

Inside brackets, multiplication takes precedence over addition.

5

Perform final operations outside brackets using standard operator precedence

45 - 14.5 \times 2 = 45 - 29 = 16

Multiplication must be executed before subtraction.

Key Concept

Hierarchy of operations in VBODMAS (Vinculum, Brackets, Orders, Division, Multiplication, Addition, Subtraction)

Question 816Question →

A standard analog clock indicates the time is exactly 4:10. Calculate the smaller angle, in degrees, formed between the hour hand and the minute hand.

Show answer & explanation

Answer: 65; 65°; 65 degrees

Answer

65 degrees

At exactly 4:10, the minute hand has moved

60^\circ

from the top of the clock, and the hour hand has moved

120^\circ

for the four hours plus an extra

5^\circ

for the ten minutes passed. The difference between their specific positions (

125^\circ

and

60^\circ

) is exactly

65^\circ

.

Step-by-Step Solution

1

Calculate the angular position of the minute hand relative to the 12 o'clock mark.

The minute hand is at 10 minutes, which corresponds to an angle of

10 \times 6^\circ = 60^\circ

.

The minute hand moves

360^\circ

in 60 minutes, which equals a speed of

6^\circ

per minute.

2

Calculate the angular position of the hour hand relative to the 12 o'clock mark.

The hour hand has moved past the 4 o'clock mark. Its position is

4 \times 30^\circ + 10 \times 0.5^\circ = 120^\circ + 5^\circ = 125^\circ

.

The hour hand moves

30^\circ

per hour and an additional

0.5^\circ

per minute due to continuous drift.

3

Find the absolute difference between the two angular positions to determine the angle between the hands.

|125^\circ - 60^\circ| = 65^\circ

.

The angle between the two hands is simply the absolute difference of their individual positions from the 12 o'clock reference point.

Key Concept

Clock hand angular positioning and the calculation of hour-hand drift.

Question 817Question →

As the Municipal Commissioner overseeing a high-value infrastructure tender, you are informed that the technical evaluation committee has shortlisted three firms. Just two days before the final financial bids are to be opened, you receive a credible anonymous email with documentary evidence showing that the Chief Engineer, who heads the evaluation committee, has a spouse holding significant undisclosed shares in the highest-rated firm. The project is already delayed by a year, and the allocated central funds will irrevocably lapse in 15 days if the contract is not awarded. What is the most administratively sound and ethically appropriate course of action?

Cancel the entire tender process immediately, suspend the Chief Engineer pending a vigilance inquiry, and permanently blacklist the highest-rated firm to set a precedent against corruption.

Proceed with the financial bid opening to ensure the project funds do not lapse, but quietly remove the Chief Engineer from the subsequent project execution phase to prevent any future undue influence.

Pause the financial bid opening, immediately constitute an independent fact-finding committee to verify the evidence within a strict timeframe, and officially apprise the funding agency of the situation to seek a provisional extension.

Privately confront the Chief Engineer with the evidence and ask him to voluntarily recuse himself from the committee retrospectively, allowing the second-highest rated firm to be awarded the contract.

Show answer & explanation

Answer: Pause the financial bid opening, immediately constitute an independent fact-finding committee to verify the evidence within a strict timeframe, and officially apprise the funding agency of the situation to seek a provisional extension.

Answer

Pause the financial bid opening, immediately constitute an independent fact-finding committee to verify the evidence within a strict timeframe, and officially apprise the funding agency of the situation to seek a provisional extension.

The correct course of action prioritizes institutional integrity and transparency. By pausing the process and initiating a time-bound inquiry, the administrator ensures due process is followed. By officially communicating with the funding agency, they handle the timeline pressure professionally without compromising ethical standards.

Step-by-Step Solution

1

Analyze the core conflict presented in the scenario.

The tension is between upholding procedural integrity (addressing the conflict of interest) and meeting strict administrative deadlines (preventing the lapse of funds).

Identifying the competing values is essential for evaluating which course of action respects the hierarchy of administrative ethics.

2

Evaluate the legality and proportionality of immediate punitive actions.

Taking drastic measures like immediate suspension or blacklisting without formal verification violates natural justice.

Administrative decisions must be backed by formal inquiries, not just initial allegations or anonymous tips.

3

Assess the transparency and formal correctness of the proposed solutions.

The correct action must involve a formal halt, a time-bound objective investigation, and transparent communication with stakeholders (the funding agency).

Transparency and adherence to due process are paramount in public administration, taking precedence over informal compromises or rushed expediency.

Key Concept

Conflict of Interest Management and Due Process in Public Procurement

Estimated Time:2m 0s

Question 818Question →

Let $N = 2^5 \times 3^4 \times 5^2$ . How many positive integer factors of $N$ are divisible by $12$ but not divisible by $15$ ?

16

32

48

24

Show answer & explanation

Answer: 16

Answer

16 positive factors

The number

N

is given by

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

. A factor

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

is divisible by

12 = 2^2 \times 3^1

if

a \in \{2,3,4,5\}

(4 choices),

b \in \{1,2,3,4\}

(4 choices), and

c \in \{0,1,2\}

(3 choices), giving

4 \times 4 \times 3 = 48

factors. Among these, those also divisible by

15 = 3^1 \times 5^1

must be divisible by

\text{LCM}(12,15) = 60 = 2^2 \times 3^1 \times 5^1

, which restricts

c \in \{1,2\}

(2 choices), yielding

4 \times 4 \times 2 = 32

factors. Subtracting these gives

48 - 32 = 16

factors.

Step-by-Step Solution

1

Express

N

in prime factorized form and define the general structure of its factors.

Any positive factor of

N = 2^5 \times 3^4 \times 5^2

has the form

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

, where

0 \le a \le 5

,

0 \le b \le 4

, and

0 \le c \le 2

.

Prime factorization determines all possible divisors of a composite number.

2

Calculate the total number of factors divisible by

12 = 2^2 \times 3^1

.

For

k

to be divisible by 12, we require

a \ge 2

(

a \in \{2,3,4,5\}

, 4 choices),

b \ge 1

(

b \in \{1,2,3,4\}

, 4 choices), and

c \ge 0

(

c \in \{0,1,2\}

, 3 choices). Total factors =

4 \times 4 \times 3 = 48

.

A factor contains another number as a divisor if all prime exponent lower bounds of the divisor are satisfied.

3

Calculate the number of factors divisible by both

12

and

15

, which is equivalent to being divisible by

\text{LCM}(12, 15) = 60 = 2^2 \times 3^1 \times 5^1

.

For

k

to be divisible by 60, we require

a \ge 2

(4 choices),

b \ge 1

(4 choices), and

c \ge 1

(

c \in \{1,2\}

, 2 choices). Total factors =

4 \times 4 \times 2 = 32

.

To exclude factors divisible by 15, we must find the intersection of multiples of 12 and multiples of 15.

4

Subtract the factors divisible by 60 from the factors divisible by 12.

Factors divisible by 12 but not by 15 =

48 - 32 = 16

.

Applying the principle of set difference yields the exact count satisfying both conditions.

Key Concept

Counting Factors with Prime Factorization Constraints

Estimated Time:2m 0s

Question 819Question →

For the composite numerical expression $E = (412^{120} + 317^{201}) - 814^{102}$ , which of the following represents its unit digit?

7

2

3

9

Show answer & explanation

Answer: 7

Answer

7

Evaluating each term by cyclicity yields unit digits of 6 for

412^{120}

, 7 for

317^{201}

, and 6 for

814^{102}

. Combining these according to the operations gives

(6 + 7) - 6 = 7

, resulting in a unit digit of 7.

Step-by-Step Solution

1

Determine the unit digit of

412^{120}

Unit digit is 6

The unit digit of base 412 is 2, which has a cyclicity of 4 (2, 4, 8, 6). The exponent 120 is divisible by 4 (

120 \pmod 4 = 0

), so we take the 4th power in the cycle:

2^4 = 16

, giving a unit digit of 6.

2

Determine the unit digit of

317^{201}

Unit digit is 7

The unit digit of base 317 is 7, which has a cyclicity of 4 (7, 9, 3, 1). The exponent remainder is

201 \pmod 4 = 1

, corresponding to

7^1 = 7

.

3

Determine the unit digit of

814^{102}

Unit digit is 6

The unit digit of base 814 is 4, which has a cyclicity of 2 (4 for odd powers, 6 for even powers). Since 102 is an even exponent, the unit digit is 6.

4

Combine the unit digits according to the expression

E = (412^{120} + 317^{201}) - 814^{102}

Unit digit is 7

Adding the unit digits of the first two terms gives

6 + 7 = 13

, which has a unit digit of 3. Subtracting 6 gives

3 - 6 = -3

, which in modulo 10 arithmetic yields

13 - 6 = 7

.

Key Concept

Cyclicity of numbers and modular arithmetic for unit digit determination

Estimated Time:1m 30s

Question 820Question →

The table below shows the annual wheat production (in metric tonnes) across four agricultural zones from 2021 to 2023:

Agricultural Zone	2021	2022	2023
Zone A	150	180	210
Zone B	200	250	300
Zone C	100	120	150
Zone D	300	330	360

Which of the following statements regarding the data given in the table are correct?

Select all that apply

Zone A registered a

20\%

growth in wheat production from 2021 to 2022.

In 2023, the wheat production of Zone B was exactly double that of Zone C.

The percentage increase in wheat production for Zone D from 2021 to 2022 was

15\%

.

Zone C produced a total of

400

metric tonnes of wheat over the three-year period from 2021 to 2023.

Show answer & explanation

Answer: Zone A registered a $20\%$ growth in wheat production from 2021 to 2022.; In 2023, the wheat production of Zone B was exactly double that of Zone C.

Answer

The correct statements are that Zone A registered a

20\%

growth in wheat production from 2021 to 2022, and in 2023, the wheat production of Zone B was exactly double that of Zone C.

Zone A's production grew from

150

to

180

tonnes, which is a growth of

\frac{30}{150} \times 100\% = 20\%

. In 2023, Zone B produced

300

tonnes while Zone C produced

150

tonnes, making Zone B's production exactly double that of Zone C.

Step-by-Step Solution

1

Calculate the percentage growth for Zone A from 2021 to 2022.

Percentage increase =

\frac{180 - 150}{150} \times 100\% = 20\%

. Statement is correct.

To verify the growth statement for Zone A using the base year 2021 value (

150

).

2

Compare Zone B and Zone C production figures for the year 2023.

Zone B produced

300

tonnes and Zone C produced

150

tonnes. Ratio =

\frac{300}{150} = 2

. Statement is correct.

To check if Zone B production was double that of Zone C in 2023.

3

Calculate the percentage increase for Zone D from 2021 to 2022.

Percentage increase =

\frac{330 - 300}{300} \times 100\% = 10\% \neq 15\%

. Statement is incorrect.

To evaluate the claim regarding Zone D's percentage growth.

4

Calculate the total wheat production of Zone C across 2021, 2022, and 2023.

Total =

100 + 120 + 150 = 370

tonnes

\neq 400

tonnes. Statement is incorrect.

To verify the three-year total production claim for Zone C.

Key Concept

Tabular Data Interpretation and Percentage Growth Analysis

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

Answer

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Common Mistakes

Answer

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Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

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Common Mistakes

Answer

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Common Mistakes

Answer

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Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

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Common Mistakes

Answer

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Common Mistakes

Answer

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Common Mistakes

Answer

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Common Mistakes

Answer

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Common Mistakes

Answer

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Common Mistakes

Answer

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Key Concept