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Pairs of numbers

I have before me four two-digit numbers, all eight digits being different. I have grouped them into pairs, such that the product of the two numbers in each pair is the same, and is a perfect cube.

What are the four numbers?

WIN £15 will be awarded to the sender of the first correct answer opened on Wednesday 14 September. The Editor’s decision is final. Please send entries to Enigma 1659, New ÐÓ°ÉÔ­´´, Lacon House, 84 Theobald’s Road, London WC1X 8NS, or to enigma@newscientist.com (please include your postal address).

Answer to 1653 Cut-free: 27 dominoes

The winner Ian Fletcher of Saxmundham, Suffolk, UK