Find the Divisibility no.
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Q. If 2³¹ is divided by 5, what will be the remainder ?
A) 1 B) 2 C) 3 D) 4
Soln:
Remainder pattern of powers of 2 when divided by 5: 2¹ → 2 , 2² → 4 , 2³ → 3 , 2⁴ → 1
Pattern = 2, 4, 3, 1 (repeats every 4 terms)
Now find where 2³¹ falls in the cycle: 31 ÷ 4 → remainder = 3
So 2³¹ is the 3rd term of the cycle.
3rd term is: 2, 4, 3, 1 → 3
Ans: 3
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Q. 7^12 - 4^12 is divisible by ?
A) 34 B) 33 C) 36 D) 35
Soln:
For even powers: a^n−b^n is divisible by (a−b)
Here: 7−4 = 3
So the whole number is divisible by 3.
Now check which option is a multiple of 3:
- A) 34 , D) 35 - No
- 33 - (multiple of 3)
- 36 - (multiple of 3)
But among these, 33 is the correct divisor for this type of expression.
Ans: 33
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Q. 2¹⁶ – 1 is divisible by which number ?
Options: A) 11 B) 13 C) 19 D) 17
Calculate the value: 2^16 = 65536
2^16−1 = 65535
Check divisibility
Try dividing by 17: 65535÷17=3855(exactly divisible)
So 17 is a factor.
Check others divisible by: 11, 13, 19
- Not divisible by 11
- Not divisible by 13
- Not divisible by 19
Ans: 17
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Q. (2⁵¹ + 2⁵² + 2⁵³ + 2⁵⁴ + 2⁵⁵) is divisible by ?
A) 127 B) 124 C) 58 D) 23
Soln:
= 2^51(1+2+4+8+16)
= 1+2+4+8+16 = 31
So the number = 2^51×31
Now check options:
- 124 = 4 × 31 → Yes (we have 31 and plenty of 2’s)
- Others do NOT contain 31.
Ans: 124
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Q: The number 5769116 is divisible by which of the following ?
A) 1 B) 5 C) 12 D) 8
Soln:
- Every number is divisible by 1
- Last digit is 6, so not divisible by 5.
- Digit sum = 35, not divisible by 3 → cannot be 12.
- Last three digits 116 not divisible by 8.
Ans: A) 1
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Q. 2918245 is divisible by which number ?
Options: A) 11 B) 12 C) 9 D) 3
Soln:
Divisibility by 11
- Rule: Difference of sums of odd and even position digits divisible by 11.
- Odd positions: 2+1+2+5 = 10
- Even positions: 9+8+4 = 21
- Difference = 10 − 21 = −11 → divisible by 11
- Not divisible by B)12 , C) 9 , D) 3 Check on Divisibility Rules
Ans: A) 11
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Q. (67⁶⁷ + 67) ÷ 68 → Remainder ?
A) 1 B) 63 C) 66 D) 67
Soln:
Short Trick:
- 67 is 1 less than 68
⇒ 67 = -1 (mod 68) - So: 67⁶⁷ + 67 = (-1)⁶⁷ + (-1)
Odd power → (–1) - So: (-1) + (-1) = -2
Convert remainder: 68 - 2 = 66
Ans = 66Divisibility Rules - Click Here
Q. 3^25+3^26+3^27+3^28 is divisible by which number ?
Options: A) 11 B) 16 C) 25 D) 30
(The expression is clearly a power series of 3.)
Soln:
= 3^25(1+3+9+27)
= 3^25 × 40
Check factors
- 40 = 2 × 2 × 2 × 5
- 3²⁵ = 3’s powers
So complete number contains: 2, 3 , 5
That means it is divisible by: 2×3×5=30
Ans: 30
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