Problem

Source: XIII International Festival of Young Mathematicians Sozopol 2024, Theme for 10-12 grade

Tags: number theory



Let \( p \neq 3 \) be a prime number. Prove that there exist natural numbers \( a \), \( b \), \( c \), \( d \), none of which are divisible by \( p \), such that \( a^2 + 3b^5 + 5c^6 + 7d^7 \) is divisible by \( p^{1000} \).