Prime Factors of 35, 36, 4734 and 4735

Theorem

The integers:

$35, 4374$

have the same prime factors between them as the integers:

$36, 4375$

Proof

We have:

\(\ds 35\)

\(=\)

\(\ds 5 \times 7\)

\(\ds 4374\)

\(=\)

\(\ds 2 \times 3^7\)

\(\ds 36\)

\(=\)

\(\ds 2^2 \times 3^2\)

\(\ds 4375\)

\(=\)

\(\ds 5^4 \times 7\)

Thus both pairs of integers can be seen to have the same prime factors:

$2, 3, 5, 7$

$\blacksquare$

Sources

• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $35$

but there is a mistake in his exposition: the pairings are the wrong way round.