Is a Number Divisible by 40

Question

One of the "shortcuts" for determining if a number is divisible by 8 is to see if the last three digits are divisible by 8. One of the "shortcuts" for determining if a number is divisible by 5 is to see whether the last digit is a 5 or a 0. If I have a number of arbitrary length, is it acceptable to say that the number is divisible by 40 if the number passes both shortcuts?

Answer 1

Yes, since $8$ and $5$ are coprime $(\gcd(8, 5)=1)$. This means that such an integer can be written as $8\times 5\times n=40\times n$, for some integer $n$.

Answer 2

To offer some clarity towards Henrik's point:

It is true that if a|n and b|n then if gcd(a,b)=1 then (ab)|n.

However, we may generalize somewhat, and note that lcm(a,b) = (ab)/gcd(a,b) will always divide n, provided that both a|n and b|n

Answer 3

Yes. Any number whose last $3$ digits are a multiple of $40$ are a multiple of $40$. It also works if the $100$s digit is even and the last $2$ digits are $00$, $40$ or $80$, or if the $100$s digit is odd and the last $2$ digits are $20$ or $60$.