Question

If 13X-9 is divisible by 25, then what will X be?
Steps would be appreciated.

Answer 1

No.. it is a question of set theory using congruence

Answer 2

I got the answer by brute force...now trying to work backwards to see what i missed.

I am assuming the answer must be an integer because otherwise it's simplistic to say X=34/13

Answer 3

it is an Integer

Answer 4

Ok...I finally used this site: http://home.scarlet.be/~ping1339/congr.htm#The-equation-ax-=-b-

Here's my initial approach:
Since we want 13X-9 to be divisible by 25, we have 13X-9 ~= 0 mod 25 (where ~= denotes congruence)


Initial simplification takes us to 13X ~= 9 mod 25, but (13,25)=1 so we can't simplify further.

Here's where I used the site. It said to look at the multiples of 13 modulo 25. I started doing this and saw that the order of them was 13, 1, 14, 2, 15, 3, ...

So if we want the multiple modulo 25 to equal 9, it'll be the (2*9)th one.

So X=18 satisfies the equation.

Not as elegant as I'd like, but it works.

Answer 5

yea it is correct....but I cant get a thing.maybe i should use the site u said..

Answer 6

It took it a while for what the site was sayingt o sink in...if you get confused by it, believe me, I understand.

But what it comes down to is this. If the greatest common divisior between the coefficient of the X (denoted a on the site) and the modulo (denoted m on the site) is 1, then taking the first (m-1) multiples of a and looking at them modulo m, you'll get every possible remainder when dividing by m...but in arbitrary order.

So for us, since 13 and 25 have no common factor, we have to take the first 24 multiples of 13 and look at the modulo 25. That gives us

13 mod 25 = 13
26 mod 25 = 1
39 mod 25 = 14
52 mod 25 = 2
65 mod 25 = 15...

and at this point we see the pattern. Since we're constructing (with this table) the value of b in 13X ~= b mod 25, to solve the question, we need b=9.

Based on the pattern, we see that we have to go to the 18th multiple to get b=9.

Any clearer?

Answer 7

yea thanks!

Answer 8

my pleasure.