Number Theory Homework: Find 3 consecutive integers...

Question

I have this problem assigned for homework, and I'm a bit confused as to how to solve it:

Obtain three consecutive integers, the first of which is divisible by a square, the second by a cube, and the third by a fourth power (other than $1^2, 1^3, 1^4$).

I've started like this, but haven't been able to get very far:

$x\equiv 0\pmod{a^2}$,

$x+1\equiv 0\pmod{b^3}$,

$x+2\equiv 0\pmod{c^4}$, some $a,b,c\in\mathbb{N}$.

Thus, we have:

$x\equiv 0\pmod{a^2}$,

$x\equiv -1\pmod{b^3}$,

$x\equiv -2\pmod{c^4}$.

I tried using the Chinese Remainder Theorem at this point but I was having difficulty considering everything is in terms of $a,b,c$...

The book's answer is $5^2\mid 350, 3^3\mid 351, 2^4\mid 352$.

Thanks!

Answer 1

HINT : Try to find an integral value say Y such that $2^2 | Y$, $3^3 | (Y + 1)$ , $4^4|(Y + 2)$ Then obtain 3 congruence relations and apply "Chinese Remainder Theorem".