Question 1047141
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Something's wrong.  The first two sentences talk only about n,
never mentioning k.  Then the last sentence asks about k, and
doesn't even mention n.  I'll try to make something of it anyway.

I think the problem should have been:</pre>When a positive integer n is divided by 5, the 
remainder is 1.  When n is divided 7, the remainder 
is 3.  What is the smallest possible positive integer n? <pre>If when n is divided by 5, the remainder is 1,
then it is 1 more than a multiple of 5. So
there is a positive integer p such that n = 5p+1

If when n is divided by 7, the remainder is 3,
then it is 3 more than a multiple of 7.  So
there is a positive integer q such that n = 7q+3

So n = 5p+1 = 7q+3

5p = 7q+2
5p = 5q+2q+2
 p = q + (2/5)q + 2/5

p-q = (2/5)q + 2/5

The left side is an integer so the right side is too.
Let that integer be A

p-q = (2/5)q + 2/5 = A

(2/5)q + 2/5 = A

2q + 2 = 5A

2q + 2 = 4A + A

q + 1 = 2A + A/2

q + 1 - 2A = A/2

The left side is an integer so the right side is too.
Let that integer be B

q + 1 - 2A = A/2 = B

A/2 = B

A = 2B

q + 1 - 2A = B

q + 1 - 2(2B) = B

q + 1 - 4B = B

q + 1 = 5B

q = 5B-1

p-q = A

p - (5B-1) = 2B

p - 5B + 1 = 2B

p = 7B-1

n = 5p+1 = 7q+3

n = 5(7B-1)+1 = 7(5B-1)+3

n = 35B-5+1 = 35B-7+3

n = 35B-4 = 35B-4

4 is less than either 5 or 7, so let C = B-1, B = C+1

n = 35(C+1)-4
n = 35C+35-4
n = 35C+31

Since 35C divided by 5 or 7 there is 0 remainder. Therefore,

1. When we divide n by 5 we get the same remainder as when
we divide 31 by 5, which gives a remainder of 1. 

2. When we divide n by 7 we get the same remainder as when
we divide 31 by 7, which gives a remainder of 3. 

So the smallest integer that when divided by 5, the remainder 
is 1, and when divided by 7, the remainder is 3 is when C=0
or n = 31. 

That's what I think was the intended answer. 

It would be impossible for n ever to be a multiple of 35, so
the misprint (typo) could not have been that k was supposed
to be n.

So the last sentence was probably from another problem, and
somehow it got placed on the wrong problem.

Edwin</pre>