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and 10 9. Use Euler's theorem to evaluate 2100000 (mod 77). An11

and 10 9. Use Euler's theorem to evaluate 2100000 (mod 77). An11

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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7.3#9

ar GENERALIZATION OF FERMAT’S THEOREM 141 relatively prime positive integers, prove that 5. If m and n are (u)pU + =1 (mod mn) (u)U be a prime Psor of n and gcd(a, p) = 1. By Fermat's theorem, aP-1 = 1 (mod p), so that aP- %3D tn for some t. Therefore aP\p-1) = (1 + tp)P = 1+ (? )(tp) + i n2) and, by induction, aP (P-1) = 1 (mod pk), where k = 1, 2, .. + (tp)P = 1 // .. Raise both des of this congruence to the $(n)/pk-l(p – 1) power to get a(n) = 1 (mod pk). Thus, %3D a$(n) = 1 (mod n). Piad the units digit of 3t0 by means of Euler's theorem. O (a) If gcd(a, n) = 1, show that the linear congruence ax = b (mod n) has the solution I= ba(n)-1 (mod n). (h) Use part (a) to solve the linear congruences 3x = 5 (mod 26), 13x = 2 (mod 40), and 10x = 21 (mod 49). 9. Use Euler's theorem to evaluate 2100000 10. For any integer a, show that a and an+l have the same last digit, 11. For any prime p, establish each of the assertions below: (a) t(p!)= 2t((p – 1)!). (b) o(p!) = (p + 1)o((p – 1)!). (c) $(p!) = (p – 1)ø((p – 1)!). 12. Given n > 1, a set of ø(n) integers that are relatively prime to n and that are incongruent modulo n is called a reduced set of residues modulo n (that is, a reduced set of residues are those members of a complete set of residues modulo n that are relatively prime to n). Verify the following: (a) The integers -31, -16, -8, 13, 25, 80 form a reduced set of residues modulo 9. (b) The integers 3, 32, 33, 34, 35, 36 form a reduced set of residues modulo 14. (c) The integers 2, 22, 2',... 13. If p is an odd prime, show that the integers %3D (mod 77). %3D %3D ,218 form a reduced set of residues modulo 27. p-1 p-1 -2, -1, 1, 2, 2. 2. form a reduced set of residues modulo p. 1.4 SOME PROPERTIES OF THE PHI-FUNCTION The next theorem points out a curious feature of the phi-function; namely, that the um of the values of ø(d), as d ranges over the positive divisors of n, is equal to n Itself. This was first noticed by Gauss. Theorem 7.6 Gauss. For each positive integer n 2 1, (p)ø 3 = u|P die sum being extended over all positive divisors of n. Positive divisor of n, we put the integer m in the class Sa provided that gcd(m, n) = d. Stated in symbols T roof. The integers between 1 and n can be separated into classes as follows: If d is a
Transcribed Image Text:ar GENERALIZATION OF FERMAT’S THEOREM 141 relatively prime positive integers, prove that 5. If m and n are (u)pU + =1 (mod mn) (u)U be a prime Psor of n and gcd(a, p) = 1. By Fermat's theorem, aP-1 = 1 (mod p), so that aP- %3D tn for some t. Therefore aP\p-1) = (1 + tp)P = 1+ (? )(tp) + i n2) and, by induction, aP (P-1) = 1 (mod pk), where k = 1, 2, .. + (tp)P = 1 // .. Raise both des of this congruence to the $(n)/pk-l(p – 1) power to get a(n) = 1 (mod pk). Thus, %3D a$(n) = 1 (mod n). Piad the units digit of 3t0 by means of Euler's theorem. O (a) If gcd(a, n) = 1, show that the linear congruence ax = b (mod n) has the solution I= ba(n)-1 (mod n). (h) Use part (a) to solve the linear congruences 3x = 5 (mod 26), 13x = 2 (mod 40), and 10x = 21 (mod 49). 9. Use Euler's theorem to evaluate 2100000 10. For any integer a, show that a and an+l have the same last digit, 11. For any prime p, establish each of the assertions below: (a) t(p!)= 2t((p – 1)!). (b) o(p!) = (p + 1)o((p – 1)!). (c) $(p!) = (p – 1)ø((p – 1)!). 12. Given n > 1, a set of ø(n) integers that are relatively prime to n and that are incongruent modulo n is called a reduced set of residues modulo n (that is, a reduced set of residues are those members of a complete set of residues modulo n that are relatively prime to n). Verify the following: (a) The integers -31, -16, -8, 13, 25, 80 form a reduced set of residues modulo 9. (b) The integers 3, 32, 33, 34, 35, 36 form a reduced set of residues modulo 14. (c) The integers 2, 22, 2',... 13. If p is an odd prime, show that the integers %3D (mod 77). %3D %3D ,218 form a reduced set of residues modulo 27. p-1 p-1 -2, -1, 1, 2, 2. 2. form a reduced set of residues modulo p. 1.4 SOME PROPERTIES OF THE PHI-FUNCTION The next theorem points out a curious feature of the phi-function; namely, that the um of the values of ø(d), as d ranges over the positive divisors of n, is equal to n Itself. This was first noticed by Gauss. Theorem 7.6 Gauss. For each positive integer n 2 1, (p)ø 3 = u|P die sum being extended over all positive divisors of n. Positive divisor of n, we put the integer m in the class Sa provided that gcd(m, n) = d. Stated in symbols T roof. The integers between 1 and n can be separated into classes as follows: If d is a
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