We will discuss here about the rules of divisibility tests by 8 and 12 with the help of different types of problems.

**1.** If ‘a’ is a positive perfect square integer, then a(a - 1) is always divisible by

(a) 12

(b) multiple of 12

(c) 12 - x

(d) 24

Solution:

‘a’ is a positive perfect square integer.

Let, a = x^{2}

Now, a (a – 1) = x^{2}(x^{2} – 1)

Therefore, a(a – 1) is always divisible by 12

Answer: (a)

**Note:** x^{2}(x^{2} – 1) is always divisible by 12 for
any positive integral values of x.

** **

**2.** If m and n are
two digits of the number 653mn such that this number is divisible by 80, then
(m + n) is equal to

(a) 2

(b) 3

(c) 4

(d) 6

**Solution:**

653xy is divisible by 80

Therefore, the values of y must be 0.

Now, 53x must be divisible by 8.

Therefore, the value of x = 6

Thus, the required sum of (x + y) = (6 + 0) = 6

Answer: (d)

**Note:** The number formed by last three digits when
divisible by 8, then the number is divisible by 8.

** **

**3.** The sum of
first 45 natural numbers will be divisible by

(a) 21

(b) 23

(c) 44

(d) 46

Solution:

Number of natural numbers (n) is 45

Therefore, Sum of numbers divisible by 45 and 46 ÷ 2 = 23

Therefore, according to the given options the required number is 23.

Answer: (b)

**Note:** Sum of ‘n’ terms of natural numbers is always
divisible by {n or n/2 or (n + 1) or (n + 1)/2} and also by the factors of n or
(n + 1)

** **

**4.** How many
digits from the unit’s digit must be divisible by 32, to make the complete
number is divisible by 32?

(a) 2

(b) 4

(c) 5

(d) None of these

Solution:

32 = 2^{5}

Therefore, required number of digits is 5

Answer: (c)

**Note:** Power of ‘2’ and ‘5’ indicate the number of
digits from the unit’s digit to decide whether the number is divisible by what
number.

** **

**5.** If 4a^{3 }+ 984
= 13b7, which is divisible by 11, then find the value of (a + b)

(a) 8

(b) 9

(c) 10

(d) 11

**Solution: **

13b7 is divisible by 11

Therefore, (3 + 7) – (1 + b) = 0

Or, 10 – 1 + b = 0

Therefore, b = 9

Now, 4a^{3} + 984 = 1397

Thus, a = 9 – 8 = 1

Therefore, required values of (a + b) = (1 + 9) = 10

Answer: (c)

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