Answer: 7
Through trial and error, you can find that 7*9 = 63 so we see that 7 is the smaller number. Or you can factor 63.
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The way your teacher probably wants you to do it is through algebra
x = smaller number
x+2 = next highest number
x*(x+2) = 63
x^2+2x = 63
x^2+2x-63 = 0
At this point, we can use the quadratic formula or factor. Factoring may be easier
x^2+2x-63 = 0
(x+9)(x-7) = 0
x+9=0 or x-7 = 0
x = -9 or x = 7
Toss out x = -9 because the instructions state that the integers must be positive.
The only valid answer is x = 7.
If x = 7, then x+2 = 7+2 = 9
So,
x*(x+2) = 63
7*9 = 63
where 7 is the smaller factor.