Given that:

2X = 1 (mod Y),

2Y = 10 (mod X),

Find the least possible sum of X and Y, where X and Y are positive integers greater than 1.

47
35
18
26

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You can start with 2 cases, the 1st case minimising X and the 2nd case minimising Y.

X must be at least 11, but no possible solution works when X = 11.

When X = 12, Y = 23.

Now let's minimising Y. 2Y = 10, so Y becomes 5.

The ones digit of 2X must be either 1 or 6, and since 2X is even, 6 is the only possibility. X can be ending with 3 or 8 for multiplication, so since X is at least 11, X is minimally 13.

The least possible sum is 5 + 13 =

18.