What is the smallest number, which when multiplied by 9 gives the prod

What is the smallest number, which when multiplied by 9 gives the product having the digit 5 only in all places ?

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9528395

61728395

12345675

59382716

Answer is Right!

Answer is Wrong!

This question was previously asked in

UPSC CISF-AC-EXE – 2023

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We are looking for the smallest number, say ‘x’, such that when multiplied by 9, the product consists only of the digit 5. Let the product be P. So, 9 * x = P, where P is a number like 5, 55, 555, 5555, etc.
For P to be divisible by 9, the sum of its digits must be divisible by 9. If P consists only of the digit 5, the sum of its digits is 5 multiplied by the number of digits. For this sum to be divisible by 9, the number of digits (all 5s) must be a multiple of 9.
The smallest number of 5s that is a multiple of 9 is 9. So, the smallest possible product P is 555,555,555 (nine 5s).
Now, we find x by dividing P by 9:
x = 555,555,555 / 9
x = 61,728,395
Comparing this with the given options, option B matches this value.

– A number is divisible by 9 if the sum of its digits is divisible by 9.
– To find the smallest such number ‘x’, the product P must be the smallest number consisting only of 5s that is divisible by 9.
– The smallest number of 5s required for the sum of digits to be divisible by 9 is nine 5s (since 9 * 5 = 45, which is divisible by 9).

The calculation 555,555,555 ÷ 9 can be done using long division or by recognizing patterns. Dividing a repdigit of ‘n’ digits by 9 follows a specific pattern. For example, 111…1 (n times) divided by 9 gives 0.111…1. Here, we have nine 5s. Dividing 111,111,111 by 9 gives 12,345,679. So, 555,555,555 = 5 * 111,111,111. Therefore, 555,555,555 / 9 = 5 * (111,111,111 / 9) = 5 * 12,345,679 = 61,728,395.

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