How do we find all possible values of $a$?

Question

$$2a5b \equiv 0 \mod 15$$ How do we find all possible values of $a$?

Here I tried to divide both sides by 2 and 5 respectively

$$ab \equiv 0 \mod 15 \implies a \in \{3,6\}, b \in \{0\}$$

However, I think the way I used is so meaningless since this is equivalence, not an equation. Thus, I could not be able to take it from there.

Regards

If $$10ab\equiv 0 \pmod {15}$$, then $$2ab\equiv 0 \pmod 3$$ right? — Rhys Hughes, Jan 20 '19 at 21:00
By $2a5b$, do you mean the number with digits $2,a,5,b$ or the product $2\cdot a \cdot 5 \cdot b$? — Jakob W., Jan 20 '19 at 21:17

score 2 · Accepted Answer · answered Jan 20 '19 at 21:14

2

Hint:

If $a,b$ are digits of the number $2a5b$ as I suppose, this number must be:

divisible by $5\quad \rightarrow \quad b=5$ or $b=0$

and

divisible by $3 \quad \rightarrow \quad 2+a+5+b$ is a multiple of $3$

can you do from this?

answered Jan 20 '19 at 21:14

Emilio Novati

1 Answers1