Question

Find the smallest nonzero t for

\[t(b-a)=0 \mod(ab) \]

Answer 1

\[\frac{t}{b}=\frac{t}{a}+n\]For integer n.

Seems most probably to be lcm(a,b), but how to prove it?

Answer 2

Hang on. Let me see if I can remember my numbeer theory correctly.

Answer 3

I make no promises, however!

Answer 4

Stange that there's a discipline where, although the problems are so easy to understand, the proofs are fiendishly hard (not necessarily here, but generally).

Answer 5

lol Yep

Answer 6

ab | t(b−a)
ab | tb and ab | ta
so b|t and a|t ... so smallest t is lcm of a and b

Answer 7

lol...I was just going to type a whole bunch of crap that didn't come close to what experiment just showed. Nice job!

Answer 8

i can only pray it's correct .. :(((((

Answer 9

I forgot the symbolism, but yes. If ab divides t(a-b) , then ab divides ta and ab divides tb. That simplifies to a divides t and b divides t. That is possible provided t is the lcm(a,b).

Answer 10

Which you have shown nicely in the proof above.

Answer 11

It looks correct