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 +. + a₁10 + ao where 0 ≤ aj ≤9. Let p = 7 or 11 or 13. (i) Show that p|m p(a₂a1a0) 10 - (a5a4a3) 10 + (a8a7α6) 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 c an E-number. Thus, for example, 2 | 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.
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 +. + a₁10 + ao where 0 ≤ aj ≤9. Let p = 7 or 11 or 13. (i) Show that p|m p(a₂a1a0) 10 - (a5a4a3) 10 + (a8a7α6) 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 c an E-number. Thus, for example, 2 | 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.
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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[Number Theory] How do you solve question 1?
![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 o'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 +. + a₁10 + 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 c an E-number.
Thus, for example, 2 | 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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F25265213-4d3d-49d7-a741-31cd431e9350%2F2802ee79-7fc2-4cba-bcd1-1dc5b56d469c%2Fgjnoqe_processed.png&w=3840&q=75)
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 o'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 +. + a₁10 + 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 c an E-number.
Thus, for example, 2 | 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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