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 ninetynine?
(a) 114345
(b) 3572404
(c) 135792
(d) 913464
Solution:
Coprime 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 ninetynine, are:
(a) 1, 9
(b) 3, 7
(c) 4, 6
(d) 5, 5
Solution:
Coprime 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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