SSC CGL 2023 Divisibility Question Test 1

Solution: The correct answer is 259.

Explanation: The expression \(6^{25} + 6^{26} + 6^{27} + 6^{28}\) can be factored as follows:

\[
6^{25} (1 + 6 + 6^2 + 6^3) = 6^{25} \times 259
\]

The factor 259 confirms that the entire expression is divisible by 259.

Solution: The correct answer is 6.

Explanation: To find the smallest number that can be subtracted from 148109326 so that it becomes divisible by 8, we need to check the last three digits of the number (326). The remainder when 326 is divided by 8 is 6. Therefore, subtracting 6 from 148109326 will make it divisible by 8.

Solution: The correct answer is 99968.

Explanation: To find the largest 5-digit number exactly divisible by 88, you start with the largest 5-digit number, which is 99999, and then find the nearest number below it that is divisible by 88. Dividing 99999 by 88 gives a quotient of 1136 with a remainder. Subtracting the remainder from 99999 gives 99968, which is the largest number that is exactly divisible by 88.
Explanation: To find the largest 5-digit number exactly divisible by 88, you start with the largest 5-digit number, which is 99999, and then divide it by 88.

\[
99999 \div 88 = 1136 \text{ remainder } 31
\]

Subtracting the remainder from 99999 gives:

\[
99999 – 31 = 99968
\]

Therefore, 99968 is the largest number that is exactly divisible by 88.

Solution: The correct answer is 4.

Explanation: To determine how many of the given numbers are divisible by 132, we can break down 132 into its prime factors:

\[
132 = 2 \times 2 \times 3 \times 11
\]

So, a number must be divisible by 4, 3, and 11 to be divisible by 132.

– **Divisibility by 4:** The last two digits of the number should be divisible by 4.
– **Divisibility by 3:** The sum of the digits of the number should be divisible by 3.
– **Divisibility by 11:** The difference between the sum of the digits in odd positions and the sum of the digits in even positions should be divisible by 11.

Let’s check each number:

1. **660**:
– Last two digits: 60 (divisible by 4)
– Sum of digits: 6 + 6 + 0 = 12 (divisible by 3)
– Difference in sums (6+0) – (6) = 0 (divisible by 11)
– **Divisible by 132**

2. **754**:
– Last two digits: 54 (divisible by 4)
– Sum of digits: 7 + 5 + 4 = 16 (not divisible by 3)
– **Not divisible by 132**

3. **924**:
– Last two digits: 24 (divisible by 4)
– Sum of digits: 9 + 2 + 4 = 15 (divisible by 3)
– Difference in sums (9+4) – (2) = 11 (divisible by 11)
– **Divisible by 132**

4. **1452**:
– Last two digits: 52 (divisible by 4)
– Sum of digits: 1 + 4 + 5 + 2 = 12 (divisible by 3)
– Difference in sums (1+5) – (4+2) = 0 (divisible by 11)
– **Divisible by 132**

5. **1526**:
– Last two digits: 26 (not divisible by 4)
– **Not divisible by 132**

6. **1980**:
– Last two digits: 80 (divisible by 4)
– Sum of digits: 1 + 9 + 8 + 0 = 18 (divisible by 3)
– Difference in sums (1+8) – (9+0) = 0 (divisible by 11)
– **Divisible by 132**

7. **2045**:
– Last two digits: 45 (not divisible by 4)
– **Not divisible by 132**

8. **2170**:
– Last two digits: 70 (not divisible by 4)
– **Not divisible by 132**

Therefore, the numbers 660, 924, 1452, and 1980 are divisible by 132, so the correct answer is 4.

Solution: The correct answer is 3.

Explanation: Since the original six-digit number is divisible by 33, it is divisible by both 3 and 11 (because \(33 = 3 \times 11\)).

When 54 is added to the number:
– **Divisibility by 3:** The number 54 is divisible by 3. Adding a number divisible by 3 to another number divisible by 3 will result in a sum that is also divisible by 3.
– **Divisibility by 11:** Adding 54 to the number doesn’t necessarily guarantee divisibility by 11, because 54 itself is not divisible by 11.

Therefore, the new number formed after adding 54 will be divisible by 3 but not necessarily by any other factors. Hence, the correct answer is **3**.

Solution: The correct answer is 49512.

Explanation: For a number to be divisible by 24, it must be divisible by both 3 and 8.

– **Divisibility by 3:** The sum of the digits of the number must be divisible by 3.
– **Divisibility by 8:** The last three digits of the number must be divisible by 8.

Let’s check each option:

1. **52668**:
– Sum of digits: \(5 + 2 + 6 + 6 + 8 = 27\) (divisible by 3)
– Last three digits: 668 (not divisible by 8)
– **Not divisible by 24**

2. **49512**:
– Sum of digits: \(4 + 9 + 5 + 1 + 2 = 21\) (divisible by 3)
– Last three digits: 512 (divisible by 8)
– **Divisible by 24**

3. **64760**:
– Sum of digits: \(6 + 4 + 7 + 6 + 0 = 23\) (not divisible by 3)
– **Not divisible by 24**

4. **26968**:
– Sum of digits: \(2 + 6 + 9 + 6 + 8 = 31\) (not divisible by 3)
– **Not divisible by 24**

Therefore, only 49512 is divisible by 24.

Solution: The correct answer is 24983.

Explanation: To determine which of these numbers is divisible by 7, we need to perform division:

– **24973 ÷ 7** ≈ 3567.57 (Not divisible by 7)
– **24983 ÷ 7** = 3569 (Divisible by 7)
– **24963 ÷ 7** ≈ 3566.14 (Not divisible by 7)
– **24993 ÷ 7** ≈ 3570.43 (Not divisible by 7)

Therefore, 24983 is the only number divisible by 7.

Solution: The correct answer is 29412.

Explanation: For a number to be divisible by 36, it must be divisible by both 4 and 9.

– **Divisibility by 4:** The last two digits of the number must be divisible by 4.
– **Divisibility by 9:** The sum of the digits of the number must be divisible by 9.

Let’s check each option:

1. **47502**:
– Last two digits: 02 (not divisible by 4)
– **Not divisible by 36**

2. **29412**:
– Last two digits: 12 (divisible by 4)
– Sum of digits: \(2 + 9 + 4 + 1 + 2 = 18\) (divisible by 9)
– **Divisible by 36**

3. **54732**:
– Last two digits: 32 (divisible by 4)
– Sum of digits: \(5 + 4 + 7 + 3 + 2 = 21\) (not divisible by 9)
– **Not divisible by 36**

4. **87064**:
– Last two digits: 64 (divisible by 4)
– Sum of digits: \(8 + 7 + 0 + 6 + 4 = 25\) (not divisible by 9)
– **Not divisible by 36**

Therefore, only 29412 is divisible by 36.

Solution: The correct answer is 22.

Explanation: To find the smallest number that should be added to 888 so that it becomes divisible by 35, we first calculate the remainder when 888 is divided by 35.

\[
888 \div 35 = 25 \text{ remainder } 13
\]

Therefore, the next multiple of 35 is \(888 + (35 – 13) = 888 + 22 = 910\).

So, the smallest number that needs to be added to 888 is 22.

Solution: The correct answer is 2.

Explanation: For a number to be divisible by 18, it must be divisible by both 2 and 9.

– **Divisibility by 2:** The last digit (X) must be even.
– **Divisibility by 9:** The sum of the digits must be divisible by 9.

Let’s calculate the sum of the digits of the number 7823326867X:

\[
7 + 8 + 2 + 3 + 3 + 2 + 6 + 8 + 6 + 7 + X = 52 + X
\]

For the sum to be divisible by 9, \(52 + X\) must be divisible by 9. The nearest multiple of 9 greater than 52 is 54.

So, \(X = 54 – 52 = 2\).

Therefore, the value of X is 2.