# Tag Info

### Find all $4$ digit numbers such that sum of digits is $11$.
As stated by Stinking Bishop in the comments, you are overcounting e.g. the "number" $(x_1, x_2, x_3, x_4) = (11, 0, 0, 0)$. The problem is that your stars and bars method does not account ...
### If $ab+1 = r^2$ for $a,b,r \in \mathbb{N},$ how to show that $\gcd(2a(r+b)+1,2b(r+a)+1) = 1?$
If there is a prime $p$ such that $p$ divides both then clearly $p\ne2$ and divides their difference $$p\mid 2r(a-b)$$ So we have 2 possibilities: If $p\mid r$ then $p\mid 2ab+1$ and $p\mid ab+1$ so \$...