But There Are Infinitely Many Pairs!
If and are integers, what is the smallest possible positive value of
The answer is 6.
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1 solution
Discussions for this problem are now closed
Because g cd ( 3 0 , 1 8 ) = 6 , all values produced by this expression must be a multiple of 6, so the smallest possible positive value would be 6, if is actually acheivable. By Bezout's identity , it is!
We can understand the basic logic of this identity by finding the values of x , y that actually give 6. To do this, we use the Euclidean Algorithm :
3 0 = 1 8 + 1 2 1 8 = 1 2 + 6 1 2 = 2 × 6
Thus, 6 = 1 8 − 1 2 = 1 8 − ( 3 0 − 1 8 ) = 2 ⋅ 1 8 − 3 0 .