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.
How many noughts are at the end of these giant numbers?
Prove that if a^2+b^2 is a multiple of 3 then both a and b are multiples of 3.
Published January 1997,October 2010,February 2011.
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.