Does $37$ divide $600000000000000074$

Question

Question: Does $37$ divide $600000000000000074$?

How would I go about this? Without using a calculator.

Some observations i made are:

$37$ is a factor of $74$ yet $37$ is prime.

I know I can divide this but I want to know if there are any tricks to this question.

Answer 1

Since $37$ divides $74$, the only way for $37$ to divide $600000000000000074$ is if it divides $600000000000000000$. You should be able to prove that that is not true.

Answer 2

Hint: $37 = \frac{111}{3}$.

You can check the divisibility of 6... with 111 very easily...

Answer 3

No. $37$ divides $74$ but does not divide $6\cdot10^n$.

Answer 4

This is an example of divide and conquer. But here it really does not matter all that much that $37$ is prime (which it is, though). The most salient point is that $37 \times 2 = 74$.

So then $600000000000000074 - 74 = 600000000000000000$. If $600000000000000074$ is divisible by $37$, then so is $600000000000000000$.

Verify that $60 = 2^2 \times 3 \times 5$ and $600 = 2^3 \times 3 \times 5^2$. Then the factorization of $6$ followed by $n$ zeroes is $2^{n + 1} \times 3 \times 5^n$. This means that $6$ followed by $n$ zeroes is divisible by $2$, $3$ and $5$. But $37$ is not divisible by any of those, and therefore can't be a divisor of $6$ followed by $n$ zeroes.

Just to be certain, check with the calculator on your computer that $600000000000000074 - 8 = 600000000000000066$ and $600000000000000066$ divided by $37$ is $16216216216216218$.