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Divisibility Question

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Divisibility Question [#permalink]

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Could you pls help me to solve one doubt. It will be great help. I do not have question in formal format. But it is something like below.

Question

Variable X is positive integer. When X is divided by 4, 6, 7 & 9, in each case remainder is 3. When X is divided by 13 remainder is 0. Find smallest value of X.

Answer is 507.



What does "when X is divided by 4, 6, 7 & 9, in each case remainder is 3" imply?
It means, if we reduce X by 3, we should have a number which is divisible by all - 4, 6, 7 and 9.
Since we are looking for the smallest such number, let's find the LCM of these numbers.

LCM of 4, 6, 7 and 9 is 4*7*9 = 252

252 and all its multiples are completely divisible by 4, 6, 7 and 9. If we add 3 to 252 or any of its multiples, they will always leave a remainder of 3 when divided by 4, 6, 7 or 9.

So X should be of the form 252n + 3.
If n = 1, X = 255. But 255 is not divisible by 13.
If n = 2, X = 507. 507 is divisible by 13 so it is the smallest such number.

For more on divisibility and questions with common remainders, check:
http://www.veritasprep.com/blog/2011/04 ... unraveled/
http://www.veritasprep.com/blog/2011/04 ... y-applied/
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Re: Divisibility Question [#permalink]

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New post 01 Oct 2017, 06:45
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Re: Divisibility Question   [#permalink] 01 Oct 2017, 06:45
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