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# When positive integer n is divided by 5, the remainder is 1.

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Intern
Joined: 07 Jun 2015
Posts: 2
Re: When positive integer n is divided by 5, the remainder is 1. [#permalink]

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14 Jan 2018, 01:35
Bunuel wrote:
bingzhang wrote:
Bunuel wrote:
SOLUTION

When positive integer n is divided by 5, the remainder is 1. When n is divided by 7, the remainder is 3. What is the smallest positive integer k such that k + n is a multiple of 35 ?

(A) 3
(B) 4
(C) 12
(D) 32
(E) 35

Positive integer n is divided by 5, the remainder is 1 --> $$n=5q+1$$, where $$q$$ is the quotient --> 1, 6, 11, 16, 21, 26, 31, ...
Positive integer n is divided by 7, the remainder is 3 --> $$n=7p+3$$, where $$p$$ is the quotient --> 3, 10, 17, 24, 31, ....

There is a way to derive general formula for $$n$$ (of a type $$n=mx+r$$, where $$x$$ is divisor and $$r$$ is a remainder) based on above two statements:

Divisor $$x$$ would be the least common multiple of above two divisors 5 and 7, hence $$x=35$$.

Remainder $$r$$ would be the first common integer in above two patterns, hence $$r=31$$.

Therefore general formula based on both statements is $$n=35m+31$$. Thus the smallest positive integer k such that k+n is a multiple of 35 is 4 --> $$n+4=35k+31+4=35(k+1)$$.

Hi Bunuel,

Would you please have a look at my solution? I've got the same answer yet a different path. Can you point it out what's wrong with my thought? Many thanks.

Question:
What is the smallest value of K, when N+K is the multiple of 35?

My solution process:
step 1. n+k=x(35), I was assuming that if x=1(because the hint is 'the smallest integer value'), then n+k=35, so k=35-n
step 2. how to get n? (the same process as you've stated above) n=31
step 3. k=35-n, and n=31, therefore k=4

Yes, the smallest value of n is 31, adding 4 gives a multiple of 35.

thank you! thank you, Bunuel. It's very nice of you.
Re: When positive integer n is divided by 5, the remainder is 1.   [#permalink] 14 Jan 2018, 01:35
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