\(\text{Suppose $a$ is an integer such that $0 \le a \le 14$, and $235935623_{74}-a$ is a multiple of $15$. What is $a$? }\)

Guest Jul 31, 2019

#1**+1 **

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.

Guest Jul 31, 2019

#2**+1 **

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

Melody Aug 1, 2019