Solution to Problem 6, Test 2

Suppose that two numbers n and n' has the same remainders when
divided by 10 and when divided by 16. This means that their
difference n-n' is divisible by 10 and 16, and thus, divisible by
their least common multiplier lcm(10,16) = 80. As we discussed in
class, you can find lcm(a,b) as the ratio (a*b)/gcd(a,b), in this
case (10*16)/2 = 80.

So, if you have two different numbers with the same remainders mod
10 and mod 16, the difference is divisible by 80. The length of
the whole interval [20,90] is 70, so on this interval, we cannot
have two numbers whose different is 80 or more - and therefore,
the reconstruction is unique.

The answer to this question would be different if we had a
different age interval, e.g, [0,90]. In this case, for numbers 10
and 90 on this interval, the difference is divisible by 80 and
hence, both numbers have the same remainders when divided by 10
and by 16.