Real Numbers — Chapter Test | Class 10 IMO

Q1

The number of prime factors of 360, counted with multiplicity, is:

360 = 2³ × 3² × 5, so the count is 3 + 2 + 1 = 6.

Q2

The decimal expansion of 17/8 is:

8 = 2³, so the expansion terminates: 17 ÷ 8 = 2.125.

Q3

Which of the following numbers is irrational?

√16 = 4, 0.25 and 22/7 are rational; √7 cannot be written as a ratio of integers, so it is irrational.

Q4

Find the odd one out: 31, 37, 41, 51.

31, 37 and 41 are prime, but 51 = 3 × 17 is composite.

Q5

Traffic lights change every 48 s, 72 s and 108 s. After starting together, they change together again after:

LCM(48, 72, 108) = 2⁴ × 3³ = 432 s.

Q6

The HCF of 6, 72 and 120 is:

6 = 2 × 3, 72 = 2³ × 3², 120 = 2³ × 3 × 5. The common factors give 2 × 3 = 6.

Q7

A floor is 672 cm by 396 cm. The largest square tile that paves it exactly has side:

672 = 2⁵ × 3 × 7 and 396 = 2² × 3² × 11, so HCF = 2² × 3 = 12 cm.

Q8

Two numbers have HCF 27 and LCM 162. If one number is 54, the other is:

Other number = (HCF × LCM) ÷ given = (27 × 162) ÷ 54 = 4374 ÷ 54 = 81.

Q9

If p and q are coprime, then HCF(p², q²) is:

Coprime numbers share no prime factors, so their squares share none either; the HCF is 1.

Q10

Two tankers hold 850 L and 680 L. The largest measuring can that fills each an exact number of times holds:

The capacity is HCF(850, 680) = 170 L.

Q11

A rational number in lowest terms has a terminating decimal exactly when its denominator has only the prime factors:

Only denominators of the form 2ᵐ × 5ⁿ produce terminating decimals.

Q12

Two numbers are called coprime when their HCF is:

Coprime numbers share no common factor other than 1, so their HCF is 1.

Q13

For every natural number n, 9ⁿ − 1 is always divisible by:

9ⁿ − 1 = (9 − 1)(9ⁿ⁻¹ + … + 1) = 8 × (an integer), so it is divisible by 8.

Q14

The product of a non-zero rational number and an irrational number is always:

Such a product can never be written as a ratio of integers, so it is always irrational.

Q15

The LCM of 96 and 404 is:

HCF(96, 404) = 4, so LCM = (96 × 404) ÷ 4 = 38784 ÷ 4 = 9696.