Question

Which one of the following number can divide 454545 completely ?

• A. 11
• B. 8
• C. 9
• D. 14

Hint:

Since 454545 is an odd number, it cannot be divided by even numbers like 8 or 14. Checking the sum of its digits (\(27\)) quickly reveals it is a multiple of 9.

Answer 1

The Correct Option is C

Concept: Divisibility rules allow us to check if an integer can evenly divide a target number without performing full long division.
• Rule for 9: An integer is divisible by 9 if and only if the sum of its individual digits is perfectly divisible by 9.
• Rule for 11: An integer is divisible by 11 if the absolute difference between the sum of the digits at odd positions and the sum of the digits at even positions is either 0 or a multiple of 11.
• Rule for 8: A number is divisible by 8 if its last three digits form a number divisible by 8.
• Rule for 14: A number is divisible by 14 if it is simultaneously even (divisible by 2) and divisible by 7.

Step 1: Test the rule of divisibility for 9 on the value 454545.
Let us compute the sum of all digits making up the given integer: \[ \text{Sum of digits} = 4 + 5 + 4 + 5 + 4 + 5 \] Group the identical digits together: \[ \text{Sum of digits} = (4 \times 3) + (5 \times 3) = 12 + 15 = 27 \] Now evaluate if this sum is divisible by 9: \[ 27 \div 9 = 3 \] Since the sum of the digits (27) is a multiple of 9, the entire number 454545 must be completely divisible by 9. Let us check this explicitly: \[ 454545 \div 9 = 50505 \] The remainder is exactly 0, confirming absolute divisibility.

Step 2: Cross-verify other options to confirm exclusivity.

• Testing 11: \[ \text{Sum of odd position digits } (1^{\text{st}}, 3^{\text{rd}}, 5^{\text{th}}) = 4 + 4 + 4 = 12 \] \[ \text{Sum of even position digits } (2^{\text{nd}}, 4^{\text{th}}, 6^{\text{th}}) = 5 + 5 + 5 = 15 \] \[ \text{Difference} = |12 - 15| = 3 \] Since 3 is not a multiple of 11, the number is not divisible by 11.
• Testing 8 and 14: The given integer 454545 is an odd number (it ends in 5). Therefore, it cannot be divided completely by any even number, automatically ruling out 8 and 14. Thus, Option (C) is the only correct answer.