\(\text{Suppose $a$ is an integer such that $0 \le a \le 14$, and $235935623_{74}-a$ is a multiple of $15$. What is $a$? }\)
If we convert 235935623 from base 74 to base 10 = 1,835,684,416,755,975. This number is ALREADY a multiple of 15, which means that a = 0.
I just lost a very time consuming answer. But yes that number mod 15 is 0 so a must be zero.
Here is the short version.
This is a very weirdly presented question. That number is base 74 (very strange) but 15 is base 10
Use the fact that 74 mod 15 = -1
So when mod 15 of that number is taken we get
2-3+5-9+3-5+6-2+3 = 0
In other words that number is already a multiple of 15 (base10) so a = 0