We will discuss here about the rules of divisibility tests by 9 and 11 with the help of different types of problems.
1. What least positive integral value must be given to * so that the number 7654*21 is divisible by 9?
(a) 1
(b) 2
(c) 3
(d) 4
Solution:
Sum of known digits of 7654*21 is 25. The no. Just greater than 25 which is divisible by 9 is 27.
Now, 25 + (*) = 27
Therefore, * = 2
Answer: (b)
Note: Sum of digits when divisible by 9, then the
no. is divisible by 9.
2. Which of the following numbers is exactly divisible by ninety-nine?
(a) 114345
(b) 3572404
(c) 135792
(d) 913464
Solution:
Co-prime factors of 99 are 9 and 11.
114345 is divisible by 99 because sum of digits is 18 and difference of (5 + 3 + 1) - (4 + 4 + 1) = 0
Therefore, required number is 114345.
Answer: (a)
Note: The difference of sums of the digits in odd and even places is zero or multiple of 11, then the no. is divisible by 11.
3. 4\(^{91}\) + 4\(^{92}\) + 4\(^{93}\) + 4\(^{94}\) is divisible by
(a) 17
(b) 13
(c) 11
(d) 3
Solution:
4\(^{91}\) + 4\(^{92}\) + 4\(^{93}\) + 4\(^{94}\)
= 4\(^{91}\)(4\(^{0}\) + 4\(^{1}\) + 4\(^{2}\) + 4\(^{3}\))
= 4\(^{91}\)(1 + 4 + 16 + 64)
= 4\(^{91}\) × 85
= 4\(^{91}\) × 5 × 17, which is divisible by 17
Therefore, the required number is 17
Answer: (a)
4. The digits indicated by ⨂ in 3422213⨂⨂ so that this number is divisible by ninety-nine, are:
(a) 1, 9
(b) 3, 7
(c) 4, 6
(d) 5, 5
Solution:
Co-prime factors of 99 are 9 and 11. Sum of the digits of 3422213xy is (17 + x + y)
According to the given options,
x + y = 10
And, (3 + 2 + 2 + 3 + y) - (4 + 2 + 1 + x) = 11
Or, 10 + y - 7 - x = 11
Or, y - x = 8
Now, x + y = 10 and y - x = 8
Therefore, x = 1 and y =9
Thus, the required numbers are 1, 9
Answer: (a)
5. The number (10\(^{25}\) - 7) is divisible by
(a) 3
(b) 7
(c) 11
(d) 13
Solution:
The number (10\(^{25}\) - 7) is divisible by 3.
Answer: (a)
Note: (10\(^{n}\) - 7) is always divisible by 3, for all values of n
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