Question 1 of the first ever International Maths Olympiad
Friday 24 July 2026 at 00:18
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Edited by Richard Walker, Friday 24 July 2026 at 00:21
From the first international Maths Olympiad 1959
Question 1
Prove that for every natural number the fraction
is already in lower terms and cannot be canceled further.
Solution
If we have a pair of natural numbers with then any common factor of and must also be a common factor of .
Any common factor of must be a common factor of and so of .
But the highest common factor of is and is therefore the highest common factor of the numerator and denominator of the given fraction, which is therefore already in lowest terms.
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