When an integer Q is divided by 5, it leaves 1 as a

Question

When an integer Q is divided by 5, it leaves 1 as a remainder. When Q is divided by 7, it leaves 3 as a remainder. Find the smallest positive Q. Then find the next smallest positive Q.

johnrb · Answer

Q = 5k + 1 and 7t + 3, where k & t are positive integers.

So 5k + 1 = 7t + 3

5k = 7t + 2

k = (7t + 2)/5. We want values of t such that 7t + 2 is divisible by 5. The lowest is 4; the next lowest is 9.

If t = 4, Q = 31.

If t = 9, Q = 66.

asdert · Answer

I like johnrb's method. Mine wasn;t as fancy.

Did 2 lists. First, the column on the right with multiples of 7. Then a column on the right, adding 3 to the number on the left:

7 10
14 17
21 24
 28 31 
35 38
42 45
49 52
56 59
 63 66 
70 73

Then went though it to find a mutliple of five+1. Whole thing didn't take a minute to do.

young_gun · Answer

agreed, this method is nice...how can you quickly find values of t such that 7t + 2 is divisible by 5?

ian7777 · Answer

Personally, when it comes to remainders, simpler is better. Lots of algebra looks great and sophisticated, but often is confusing and time consuming. I think this method, though not "as fancy", yields the right answer nearly immediately.

Note that asdert started with multiples of 7. That's the right move. S/He could have started with 5, but then there are just more numbers to write out on the list. So start with the higher number, then make the other number conform.

When an integer Q is divided by 5, it leaves 1 as a

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