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[NUMBER THEORY] How do you solve question 2 (ii)

[NUMBER THEORY] How do you solve question 2 (ii)

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section: Chapter Questions
Problem 63RE
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[NUMBER THEORY] How do you solve question 2 (ii)

1. Recall that for a real number x, [x] is the largest integer m with m < x. (i) Show that, for a positive integer n and a prime p, the largest power of p occurring in the prime factorization of n! is [n/p] + [n/p²] + [n/p³] +.... (ii) How many 0's are there at the end of 115!? At the end of 1151! ? Note that 10! = 3628800 ends in 2 zeros. Justify your answer. 2. (i) For some positive integer n, suppose that 2k is the largest power of 2 in the set {1,...,n}. Show that 2k does not divide any other element of this set. (ii) Show that the number 1 +/+ + is not an integer. 3. In this problem we use the notation for the digits a, of an integer m given by m = (akak-1... ao ) 10 = ak10k +ak-110k−¹ +. + a₁ 10+ ao where 0 ≤ aj ≤9. Let p = 7 or 11 or 13. (i) Show that p|m p(a₂a1a0) 10 - (a5a4a3) 10 + (а8а7α6) For example, to check if 13 | 75787192, it is enough to check if 13 divides the number 192 - 787 + 75. (ii) Is there digit x such that the number x75787192 divisible by 77? 4. In Silverman's E-world¹, the E-numbers (even numbers) are the set E = {..., -4,-2,0, 2, 4, 6,... } = 2Z with the usual operations of + and. For E-numbers a and b, we say that b|Ea⇒a=bc with e an E-number. Thus, for example, 2 E 8 but 2 E 6. A positive E-number is an E-prime if it is not divisible by any positive E-numbers. Note that 1 is not an E-number and that an E-number does not divide itself! (i) Show that every (positive) E-number can be written as a product of E-primes. (ii) Show that a € E with a > 0 is an E-prime if and only if 4 | a. (iii) Find all different E-prime factorizations of the number 840. (iv) Show that the number of different E-prime factorizations of an E-number n can be arbitrarily large.
Transcribed Image Text:1. Recall that for a real number x, [x] is the largest integer m with m < x. (i) Show that, for a positive integer n and a prime p, the largest power of p occurring in the prime factorization of n! is [n/p] + [n/p²] + [n/p³] +.... (ii) How many 0's are there at the end of 115!? At the end of 1151! ? Note that 10! = 3628800 ends in 2 zeros. Justify your answer. 2. (i) For some positive integer n, suppose that 2k is the largest power of 2 in the set {1,...,n}. Show that 2k does not divide any other element of this set. (ii) Show that the number 1 +/+ + is not an integer. 3. In this problem we use the notation for the digits a, of an integer m given by m = (akak-1... ao ) 10 = ak10k +ak-110k−¹ +. + a₁ 10+ ao where 0 ≤ aj ≤9. Let p = 7 or 11 or 13. (i) Show that p|m p(a₂a1a0) 10 - (a5a4a3) 10 + (а8а7α6) For example, to check if 13 | 75787192, it is enough to check if 13 divides the number 192 - 787 + 75. (ii) Is there digit x such that the number x75787192 divisible by 77? 4. In Silverman's E-world¹, the E-numbers (even numbers) are the set E = {..., -4,-2,0, 2, 4, 6,... } = 2Z with the usual operations of + and. For E-numbers a and b, we say that b|Ea⇒a=bc with e an E-number. Thus, for example, 2 E 8 but 2 E 6. A positive E-number is an E-prime if it is not divisible by any positive E-numbers. Note that 1 is not an E-number and that an E-number does not divide itself! (i) Show that every (positive) E-number can be written as a product of E-primes. (ii) Show that a € E with a > 0 is an E-prime if and only if 4 | a. (iii) Find all different E-prime factorizations of the number 840. (iv) Show that the number of different E-prime factorizations of an E-number n can be arbitrarily large.
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