By Chowdhury K.C.

ISBN-10: 8186299653

ISBN-13: 9788186299654

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Extra resources for A first course in theory of numbers

Example text

Sometimes we are only interested in solutions within a certain range. 54 For instance, suppose that in the above example we want to find all positive solutions. Thus we need to find all those values of t for which -39 - 30t> 0 186 + 143t > O. and The first of these inequalities implies that t second implies that t ~ -1. ~ -2 whilst the Thus in this case we see that there are in fact no positive solutions. EXERCISES I SECTION 1 1. Prove each of the following statements by induction. Try to use both methods of writing out your proof.

Deduce that for any numbers a and b, lcm(a,b) i f (a,b) 9. 1 i f and only = 1.

The positive divisors of 18 are 1,2,3,6,9,18, whilst those of -24 are 1,2,3,4,6,8,12,24. The common divisors of 18 and -24 (amongst the positive numbers) are thus 1,2,3,6. ) gcd(18,-24). = Notice that, as this example indicates, the gcd of two integers is always positive, regardless of the sign of the two given integers. A very useful result concerning the gcd of two integers is that gcd(a,b) can always be expressed as a linear combination (with integer coefficients) of a and b. (18). Let a,b be integers, not both zero.

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A first course in theory of numbers by Chowdhury K.C.


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