The least common multiple (LCM) of two integers a and b is the least number c such that an = c and bm = c for some integers n and m.
We can find the LCM of two integers by looking at their prime factorizations, and then taking the product of the least number of primes needed to "contain" both. For example, to find the least common multiple of 28 and 30, we note that
28 = 2^2*7
and
30 = 2*3*5
In order to be divisible by 28, the LCM must have 2^2 as a factor. This also takes care of the 2 in 30. In order to be divisible by 30, it must also have 5 as factor. Finally, it must have 7 as a factor, too, to be divisible by 28. Thus, the LCM of 28 and 30 is
2^2*5*7*3 = 420
If we look at the prime factorizations of 84 and 504, we have
84 = 2^2*3*7
and
504 = 2^3*3^2*7
Working backwards, we know that 2^3 must be a factor of N, or else the LCM would only need 2^2 as a factor. Similarly, we know 3^2 is a factor of N or else the LCM would only need 3 as a factor. Then, as 7, the only other factor of the LCM, is needed for 84, N may or may not have 7 as a factor. Thus, the two possibilities for N are:
N = 2^3 * 3^2 = 72
or
N = 2^3*3^2*7 = 504
