We will discuss here about the test of divisibility tests with the help of different types of problems.

**1.** Find the common multiples of 15 and 25, which is nearest to 500:

(a) 450

(b) 525

(c) 515

(d) 500

Solution:

LCM of 15 and 25 is 75.

75 × 6 = 450 and 75 × 7 =525

500 – 450 > 525 – 500

Therefore 525 is the nearest

Answer: (b)

**2.** When a certain number is multiplied by 13, the product
consists entirely of fives. The smallest such number is:

(a) 41625

(b) 42515

(c) 42735

(d) 42135

Solution:

Let the number be x

Now, 13 × x = 555555

Therefore, x = \(\frac{555555}{13}\) = 42735

Answer: (c)

**Note:** Any six-digit no. of same digit is divisible by 3, 7,
11, 13 and 37.

**3.** The greatest number, by which the product of three
consecutive multiples of 3 is always divisible, is:

(a) 54

(b) 81

(c) 162

(d) 243

Solution:

Of any three consecutive numbers, one of the numbers must be even. And , out of three consecutive multiple of 3, one no. must be multiple of 3\(^{2}\).

Therefore, required number = 3\(^{2 + 1 + 1}\) × 2 = 162

Answer: (c)

**Note:** Product of three consecutive multiple of 3 is always
divisible by 3\(^{4}\) × 2 = 81 × 2 = 162

**4.** The largest number by which the expression (n\(^{3}\) – n) is
always divisible for all positive integral values of ‘n’ is:

(a) 3

(b) 4

(c) 5

(d) 6

Solution:

The required number is 6

Answer: (d)

**Note:** If ‘n’ is a positive integer then (n\(^{3}\) - n) is always
divisible by 6 and (n\(^{5}\) - n) is always divisible by 30.

**5.** The largest number that exactly divides each term of the
sequence

1\(^{5}\) - 1, 2\(^{5}\) - 2, 3\(^{5}\) - 3, ......................., n\(^{5}\) - n is

(a) 1

(b) 15

(c) 30

(d) 120

Solution:

(n^{5} - n) is always divisible any 30, for any integral
values of ‘n’.

Answer: (c)

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