What is the smallest multiple of 77 that is greater than 70,000? 700,000?

Question

What is the smallest multiple of $77$ that is greater than $70000$? $700000$?

How would you approach this question? My first thought was to factorize the $77$ into $7$ and $11$ but then how would you choose the exponents?

Answer 1

In general you can do this by first dividing by $77$, obtaining some remainder $r$. If we then add $77-r$ to the number, we obtain the smallest such multiple.

Answer 2

We know $70000 \equiv 0 {\pmod 7}$. Next we compute its residue under mod 11.

$$ 70000=7\cdot 10^4 \equiv 7 \cdot (-1)^4\pmod{11} $$

I will leave the rest for you.

Answer 3

A multiple of $77$ is multiple of $7$ and $11$ simultaneously.

$70070$ is clearly multiple of $11$. (The two $7$'s are placed in odd and even places; the alternating sum is zero. )

It is obviously multiple of $7$. Hence answer is $\boxed{70070}$.

By exactly same logic, the other number is $\boxed{700007}$.