Let us develop a general divisibility criterion that proceeds recursively. Suppose that n is given in...
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Let us develop a general divisibility criterion that proceeds recursively. Suppose that n is given in decimal n = (akak-1...a1a0) base 10. Let d 22 be any positive integer with gcd(d, 10) = 1. Let e = N be a multiplicative inverse of 10 modulo d. Define n'=(n-a0 )/10 +e *a0. Prove that d | n if and only if d | n'. Thus, we can determine whether n is divisible by d by constructing the sequence n, n' (n')'.... until we reach a term small enough that we can tell whether it is divisible by d. Let us develop a general divisibility criterion that proceeds recursively. Suppose that n is given in decimal n = (akak-1...a1a0) base 10. Let d 22 be any positive integer with gcd(d, 10) = 1. Let e = N be a multiplicative inverse of 10 modulo d. Define n'=(n-a0 )/10 +e *a0. Prove that d | n if and only if d | n'. Thus, we can determine whether n is divisible by d by constructing the sequence n, n' (n')'.... until we reach a term small enough that we can tell whether it is divisible by d.
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Proof If If n is divisible by d then n must also be divisible by 10d Therefore we can write n as n 1... View the full answer
Related Book For
Fundamentals of Corporate Finance
ISBN: 978-0077861629
8th edition
Authors: Richard Brealey, Stewart Myers, Alan Marcus
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