How many 2 digit numbers satisfy the following congruences, x-=3(mod7), x-=2(mod5)

Answer 1

The two congruences can be rewritten as linear equations as follows:

x ≡ 3(mod 7) --> x = 3 + 7m and

x ≡ 2(mod 5) --> x = 2 + 5n

for some integers m, and n

We can then consider the possible two-digit x's that arise from each equation:

3 + 7m: 10 17 24 31 38 45 52 59 66 73 80 87 94

2 + 5n: 12 17 22 27 32 37 42 47 52 57 62 67 72 77 82 87 92 97

Our calcuations and search of the first line are simplified by noticing that we only need consider numbers ending in 2 or 7.

We thus find that 3 two-digit numbers satisfy these congruences: x = 17, x = 52, and x = 87.