Divisibility Intuition Math Example 4

Solution

1. 1
   Remainder $2$ mod $5$: $N$ ends in $2$ or $7$ (since $12, 17, 22, 27, \ldots$ all leave remainder $2$).
2. 2
   Remainder $1$ mod $3$: digit sum $\equiv 1 \pmod{3}$ (not divisible by $3$).
3. 3
   For $N$ to be divisible by $15$, it must be divisible by both $3$ and $5$. But it is not divisible by $5$ (remainder $2$) or $3$ (remainder $1$), so $N$ is definitely not divisible by $15$.