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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