Show that, if n 2 and if p is a prime which divides n but...
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Show that, if n ≥ 2 and if p is a prime which divides n but is such that p² is not a factor of n, then p(n)+1 = p mod n. Can you find and prove a generalisation of this? [Hint for first part: n may be written as pm, and (p, m) = 1; consider powers of p modulo m. Hint for second part: for example, you may check that, although 29(100)+1 is not congruent to 2¹ mod 100, one does have 2(100)+2 congruent to 2² mod 100; also 620+1 = 6 mod 66.] Show that, if n ≥ 2 and if p is a prime which divides n but is such that p² is not a factor of n, then p(n)+1 = p mod n. Can you find and prove a generalisation of this? [Hint for first part: n may be written as pm, and (p, m) = 1; consider powers of p modulo m. Hint for second part: for example, you may check that, although 29(100)+1 is not congruent to 2¹ mod 100, one does have 2(100)+2 congruent to 2² mod 100; also 620+1 = 6 mod 66.]
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Related Book For
Finite Mathematics and Its Applications
ISBN: 978-0134768632
12th edition
Authors: Larry J. Goldstein, David I. Schneider, Martha J. Siegel, Steven Hair
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