Number Theory

We are starting with this:

$N \equiv 4 \pmod{6}, \\ N \equiv 4 \pmod{10}, \\ N \equiv 4 \pmod{15}, \\ N \equiv 4 \pmod{24}.$

We can rewrite this as:

$N = 6a +4\\N=10b+4\\N=15c+4\\N=24d+4$

$N -4 = 6a\\N-4=10b\\N-4=15c\\N-4=24d$

Therefore, N - 4 is a factor of 6, 10, 15, and 24. To find the minimum value of N-4, we find the least common multiple of 6, 10, 15, 24. We can do this by listing out the factors:

$6 = 2^13^1$

$10=2^15^1$

$15=3^15^1$

$24=2^33^1$

$N-4= \text{lcm}(6, 10, 15, 24) = 2^33^15^1=120$

Therefore, the least possible value of N is 124.