Find all 3 digit numbers such that by adding the first digit, the
square of the second and the cube of the third you get the original
number, for example 1 + 3^2 + 5^3 = 135.
Factorial one hundred (written 100!) has 24 noughts when written in full and that 1000! has 249 noughts? Convince yourself that the above is true. Perhaps your methodology will help you find the number of noughts in
10 000! and 100 000! or even 1 000 000!
Prove that if a^2+b^2 is a multiple of 3 then both a and b are multiples of 3.
In step 1 of the method, we used the fact that there is a unique
number in the $7$ times table (up to $10 \times 7 = 70$) ending in
each digit $0$ to $9$. Is this true in other times tables? It works
for $3, 7,9, 11\ldots$. But it certainly doesn't work for $2$ - no
multiple of $2$ ends in $1$. It turns out that works for any number
not divisible by $2$ or $5$ (the prime factors of $10$). To
understand why this is true, read this
article on the Chinese Remainder Theorem.