i factored 14 and 4 as follows

2X7 , 2X2

I get 64 and 28

which means 65 and 29 are possible.

how do you get the 84 from the factorization?

Hey TJ,

If you break down the common multiples of 14 and 4 that way you'd recognize that you need a number divisible by 2*2*7 = 28. Any number that contains those factors will work:

28 (2*2*7)
56 (2*2*7*2)
84 (2*2*7*3)

Because they limit it to 2-digit numbers, there won't be that many possibilities, so for a question like this is may be just as easy to just write out the multiples of 14 (since there will be fewer) and add the remainder of 1 to each:

14 --> 15
28 --> 29
42 --> 43
56 --> 57
70 --> 71
84 --> 85
98 --> 99

Then check to see which of those will yield a remainder of 1 when divided by 4:

29 (7*4 + 1)
57 (14*4 +1)
85 (21*4 + 1)

So there are three numbers that meet both conditions.