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College Algebra (MindTap Course List)
College Algebra (MindTap Course List)
12th Edition
ISBN:9781305652231
Author:R. David Gustafson, Jeff Hughes
Publisher:R. David Gustafson, Jeff Hughes
Chapter8: Sequences, Series, And Probability
Section8.5: Mathematical Induction
Problem 27E
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6. Let a be a number written in base 10 as x = ao + a110 +..a, 10" where 0< a; < 10. (a) Prove that 2 divides a if and only if 2 divides ao. (b) Prove that 4 divides a if and only if 4 divides ao +2a1. (c) Prove that 8 divides a if and only if 8 divides ao + 2a1 + 4a2. (d) Prove that 5 divides a if and only if 5 divides ao. (e) Prove that 9 divides a if and only if 9 divides the sum ao + ...a, of its digits. (f) Prove that 3 divides a if and only if 3 divides the sum ao + ...an of its digits. (g) Prove that 11 divides a if and only if 11 divides the alternating sum ao – a1 + az .... (h) Come up with a rule for dividing by 7 and prove that it works.
Transcribed Image Text:6. Let a be a number written in base 10 as x = ao + a110 +..a, 10" where 0< a; < 10. (a) Prove that 2 divides a if and only if 2 divides ao. (b) Prove that 4 divides a if and only if 4 divides ao +2a1. (c) Prove that 8 divides a if and only if 8 divides ao + 2a1 + 4a2. (d) Prove that 5 divides a if and only if 5 divides ao. (e) Prove that 9 divides a if and only if 9 divides the sum ao + ...a, of its digits. (f) Prove that 3 divides a if and only if 3 divides the sum ao + ...an of its digits. (g) Prove that 11 divides a if and only if 11 divides the alternating sum ao – a1 + az .... (h) Come up with a rule for dividing by 7 and prove that it works.
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