What is the smallest positive integer other than 2 which when divided by 4 or 9 leaves a remainder of 2?

The use of the Word 'or' means that the smallest positive integer is 6, as 6 ÷ 4 leaves a remainder of 2.

If you mean the smallest positive integer what when divided by both 4 and 9 leaves a remainder of 2 then the answer is 38.

4x + 2 = M : 4x = M - 2

9y + 2 = M : 9y = M - 2

4x = 9y : x = 9y/4: x = 2.25y

This means that for x to be an integer, y must = 4 or a multiple of 4.

Thus, if y = 4 then 9y + 2 = 36 + 2 = 38

And, x = 9

If higher value integers were required then put y = 8,( or higher multiples of 4)

Then 9y + 2 = 72 + 2 = 74

And, x = 18