Question

How many positive integers less than 100 have a remainder of 2 when divided by 13?

Answer 1

\frac{X-2}{13} = n where n is a non-negative integer, and X is a positive integer less than 100. Basically determine how big the subset for possible values of n, and you have your answer.

Rearrange the above equation to:

13n + 2 = X

and since X < 100,

13n + 2 < 100
13n < 98
n < 98/13

Since n must be an integer....

n < 8

So there are 8 possible values for n (0 to 7), and therefore 8 positive integers less than 100 that have a remainder of 2 when divided by 13. Hope this helps.

Answer 2

y is ther 8 ? i got 7....

Answer 3

Hang on a second.

Answer 4

I was wrong the answers is 13. 10 is the first positive integer that can be divide by 7 with a remainder of 3. The next is 17, then 24, ... all the way to 94. So the answer is 13. Are you following me?

Answer 5

@FelicitySchimke, you had the right approach, just need to exclude zero since it is not a positive integer.

Here's just a summary using a slightly different approach.

All integers with a remainder of 2 that are positive and less then 100 can be written in set notation as:
$$
\{x=13n+2|n\in Z,0<x<100\}\\
$$
This implies that
$$
n=\{1,2,3,4,5,6,7\}
$$
Since for \(n=8\), \(x=13\times8+2=106\), which is greater than 100 and is not part of the solution; but for \(n=7\), \(x=93\).

http://www.wolframalpha.com/input/?i=x%3D13n%2B2%2C+0+%3C+n+%3C+100%2Cn+is+integer