Divisibility Tests

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:

(n⁵ - n) is always divisible any 30, for any integral values of ‘n’.

Answer: (c)

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