Problem 1

Find all pairs

(a,b)

of positive integers such that

a^3

is a multiple of

b^2

, and

b-1

is a multiple of

a-1

. (Here

n

is called a multiple of

m

n=km

for some integer

k

Step 2 of 5: Introduce c with a^3=b^2c

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a^3=b^2c,\quad a\equiv b\equiv1\pmod{a-1}\ \Longrightarrow\ c\equiv1\pmod{a-1}

Detailed analysis

Now suppose $b\ge a$ . Since $b^2$ divides $a^3$ , write $a^3=b^2c$ for a positive integer $c$ . Modulo $a-1$ we have $a\equiv1$ , and since $(a-1)\mid(b-1)$ also $b\equiv1$ , hence $b^2\equiv1$ and $a^3\equiv1\pmod{a-1}$ . From $a^3=b^2c$ this forces $c\equiv1\pmod{a-1}$ .