Number System — Practice

Question 1

How many factors does 720 have?
(a) 24 (b) 28 (c) 30 (d) 32

Accepted Answer

(c) 30.

720 = 2⁴ × 3² × 5¹. Number of factors = (4+1)(2+1)(1+1) = 5 × 3 × 2 = 30.

Question 2

The HCF of two numbers is 12 and their LCM is 336. If one number is 84, the other is:
(a) 36 (b) 48 (c) 56 (d) 72

Accepted Answer

(b) 48.

Other number = (HCF × LCM) ÷ 84 = (12 × 336) ÷ 84 = 4032 ÷ 84 = 48. Check: HCF(84, 48) = 12 ✓.

Question 3

What is the largest 4-digit number divisible by 88?
(a) 9724 (b) 9856 (c) 9944 (d) 9988

Accepted Answer

(c) 9944.

9999 ÷ 88 = 113.6…, so take 88 × 113 = 9944. Verify with coprime split 88 = 8 × 11: last three digits 944 = 8 × 118 ✓; alternating sum (4 + 9) − (4 + 9) = 0 ✓.

Question 4

The six-digit number 476xy0 is divisible by both 3 and 11. What is the maximum possible value of x + y?
(a) 7 (b) 10 (c) 13 (d) 15

Accepted Answer

(c) 13.

Rule of 3: digit sum 17 + x + y must be a multiple of 3, so x + y ≡ 1 (mod 3). Rule of 11: (0 + x + 7) − (y + 6 + 4) = x − y − 3 must be a multiple of 11, giving x − y = 3 (x − y = −8 forces x = 0, y = 8, but then x + y = 8 ≢ 1 mod 3 — rejected). With x = y + 3: pairs (5,2) and (8,5) satisfy both rules. Maximum x + y = 8 + 5 = 13.

Question 5

What is the remainder when 2⁴⁷ is divided by 7?
(a) 1 (b) 2 (c) 4 (d) 6

Accepted Answer

(c) 4.

2³ = 8 ≡ 1 (mod 7). Write 47 = 3 × 15 + 2, so 2⁴⁷ = (2³)¹⁵ × 2² ≡ 1 × 4 = 4.

Question 6

The unit digit of 7¹⁰⁵ is:
(a) 1 (b) 3 (c) 7 (d) 9

Accepted Answer

(c) 7.

Powers of 7 end in the cycle 7, 9, 3, 1 (length 4). 105 mod 4 = 1 → first entry of the cycle → 7.

Question 7

Four bells toll at intervals of 6, 8, 12 and 18 seconds. They toll together at 12:00 noon. How many more times will they toll together in the next hour?
(a) 48 (b) 50 (c) 60 (d) 72

Accepted Answer

(b) 50.

They coincide every LCM(6, 8, 12, 18) = 72 seconds. In 3600 seconds there are 3600 ÷ 72 = 50 such moments after the start.

Question 8

A number divided by 56 leaves remainder 29. What is the remainder when the same number is divided by 8?
(a) 1 (b) 3 (c) 5 (d) 7

Accepted Answer

(c) 5.

N = 56k + 29. Since 8 divides 56, N mod 8 = 29 mod 8 = 5.

Question 9

The greatest number which divides 62, 132 and 237 leaving the same remainder in each case is:
(a) 21 (b) 30 (c) 35 (d) 70

Accepted Answer

(c) 35.

The common remainder cancels in differences: 132 − 62 = 70, 237 − 132 = 105, 237 − 62 = 175. HCF(70, 105, 175) = 35. Check: 62, 132, 237 all leave remainder 27 on division by 35 ✓.

Question 10

What is the remainder when 3¹⁰⁰ is divided by 13?
(a) 1 (b) 3 (c) 9 (d) 12

Accepted Answer

(b) 3.

Hunt the cycle: 3¹ = 3, 3² = 9, 3³ = 27 ≡ 1 (mod 13). Write 100 = 3 × 33 + 1, so 3¹⁰⁰ = (3³)³³ × 3 ≡ 1 × 3 = 3.

Practice — Number System

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