Question

Prove that any integer is the sum of two relatively prime integers. (We say two integers and are relatively prime if their HCF (i.e., GCD) is )

Hard

Solution

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Let be any integer.

When is odd then where either is even or is even.

If is even then is odd and hence and are relatively prime to each other. Similarly, vice versa.

Therefore,

Hence, any integer  is the sum of two relatively prime integers.

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