SSC CGL Number System Concept Check

1. How many prime numbers are strictly between 44 and 64?

• A. 3
• B. 4 (Correct)
• C. 5
• D. 6

Explanation: The primes are 47, 53, 59 and 61. Check odd candidates for divisibility by 3, 5 and 7; these are the only primes at most the square root of 64. There are four.

2. In the four-digit number 6a28, a is a digit. Which value of a makes the number divisible by 9?

• A. 0
• B. 1
• C. 2 (Correct)
• D. 5

Explanation: The digit sum is 6 + a + 2 + 8 = 16 + a. For a from 0 to 9, the only multiple of 9 available is 18, giving a = 2. Check: 6228 = 9 × 692.

3. A workshop has strips 462 cm and 630 cm long. Both are cut completely into equal pieces of the greatest possible length. What is that length?

• A. 21 cm
• B. 42 cm (Correct)
• C. 63 cm
• D. 84 cm

Explanation: The piece length must divide both lengths. Euclid's algorithm gives 630 − 462 = 168, 462 = 168 × 2 + 126, 168 = 126 + 42, and 126 = 42 × 3. HCF = 42 cm. The strips yield 11 and 15 pieces.

4. Three display lights turn on together, then repeat every 18, 24 and 40 seconds. After how many seconds will they next turn on together?

• A. 120
• B. 180
• C. 240
• D. 360 (Correct)

Explanation: 18 = 2 × 3², 24 = 2³ × 3 and 40 = 2³ × 5. LCM = 2³ × 3² × 5 = 360 seconds. This is divisible by all three intervals.

5. What is the smallest integer greater than 500 that is divisible by both 12 and 18?

• A. 502
• B. 504 (Correct)
• C. 516
• D. 540

Explanation: LCM(12,18) = 36. Since 36 × 13 = 468 and 36 × 14 = 504, the first common multiple above 500 is 504. Being divisible by just one of the numbers is not enough.

6. What is the greatest positive divisor that leaves remainder 5 when dividing each of 89 and 137?

• A. 6
• B. 12 (Correct)
• C. 24
• D. 44

Explanation: Subtract the remainder: 89 − 5 = 84 and 137 − 5 = 132. Their HCF is 12. Since 5 < 12, it is valid: 89 = 12 × 7 + 5 and 137 = 12 × 11 + 5.

7. What remainder does 7^43 leave when divided by 5?

• A. 1
• B. 2
• C. 3 (Correct)
• D. 4

Explanation: 7 has remainder 2 modulo 5. The powers of 2 have remainders 2, 4, 3, 1 in a four-term cycle. Since 43 = 4 × 10 + 3, select the third remainder, 3.

8. What is the unit digit of 3^28 + 4^15?

• A. 3
• B. 5 (Correct)
• C. 7
• D. 9

Explanation: Powers of 3 end in 3, 9, 7, 1. As 28 is a multiple of 4, 3^28 ends in 1. Odd positive powers of 4 end in 4, so 4^15 ends in 4. Their sum ends in 5.

9. How many positive divisors, including 1 and itself, does 432 have?

• A. 12
• B. 16
• C. 20 (Correct)
• D. 24

Explanation: 432 = 16 × 27 = 2⁴ × 3³. Choose the exponent of 2 in five ways (0 to 4) and of 3 in four ways (0 to 3). Total positive divisors = 5 × 4 = 20.

10. What is the smallest positive integer k for which 1080k is a perfect square?

• A. 6
• B. 10
• C. 15
• D. 30 (Correct)

Explanation: 1080 = 2³ × 3³ × 5. All three exponents are odd, so supply one factor each of 2, 3 and 5. k = 30. Check: 1080 × 30 = 32400 = 180².

11. How many trailing zeros are there in the decimal representation of 64!?

• A. 12
• B. 13
• C. 14 (Correct)
• D. 16

Explanation: Count factors of 5: floor(64/5) = 12 and floor(64/25) = 2. Higher powers of 5 contribute zero. Total = 14. The two multiples of 25 each contribute an additional five.

12. What is the minimum number of digits needed after the decimal point to write 21/84 exactly as a terminating decimal?

• A. 1
• B. 2 (Correct)
• C. 3
• D. It does not terminate

Explanation: Reduce 21/84 to 1/4. Since 4 = 2², multiplying numerator and denominator by 25 gives 25/100 = 0.25, requiring two decimal places.

13. Integers a and b leave remainders 5 and 6, respectively, when divided by 8. What is the standard non-negative remainder when a − b is divided by 8?

• A. 0
• B. 1
• C. 6
• D. 7 (Correct)

Explanation: Subtract the remainders: 5 − 6 = −1. Add 8 to put the result in the range 0 to 7. Thus a − b = 8q + 7 for an integer q, and the remainder is 7.

14. How many unordered pairs of positive integers have HCF 8 and LCM 240?

• A. 2
• B. 3
• C. 4 (Correct)
• D. 8

Explanation: Write the numbers as 8u and 8v with co-prime u,v. Then uv = 240/8 = 30. The unordered factor pairs (1,30), (2,15), (3,10) and (5,6) are all co-prime. They give four pairs: (8,240), (16,120), (24,80) and (40,48).

15. If x = 0.363636... with the block 36 repeating forever, what is 1/x?

• A. 4/11
• B. 11/4 (Correct)
• C. 25/9
• D. 9/25

Explanation: 100x − x = 36, so x = 36/99 = 4/11. The question asks for its reciprocal, which is 11/4. Treating the repeating decimal as the finite number 0.36 would give the wrong result.