2. (i) For some positive integer n, suppose that 2 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 +/+ · + ½ 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 + (a8a7a6) .... 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 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. c) C
2. (i) For some positive integer n, suppose that 2 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 +/+ · + ½ 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 + (a8a7a6) .... 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 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. c) C
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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