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Proposition 6.1.21. A number is divisible by 11 if and only if the alternat- ing sums of the digits is divisible by 11. (Note: alternating sums is where the signs of the number alternate when summing.) PROOF. Given an integer with digits do . dn where the number is writeen as dndn-1...d1do we can write n = dm · 10" + dm-1 · 10m-1 +...+ do · 10º it follows that: mod (n, 11) = mod (dm · 10m + dm-1 · 10m-1 + ..+ do · 10º, 11) = mod (dm · (-1)™ + dm-1 · (-1)"m-1 + ... = mod ((-1)"(dm – dm-1+ ·· [substitution] + do · (–1)º, 11) [mod(10, 11) = –1] + do · 1), 11) [factor out (-1)"] Therefore, mod (n, 11)=0 if and only if the alternating sums of the digits of

Proposition 6.1.21. A number is divisible by 11 if and only if the alternat- ing sums of the digits is divisible by 11. (Note: alternating sums is where the signs of the number alternate when summing.) PROOF. Given an integer with digits do . dn where the number is writeen as dndn-1...d1do we can write n = dm · 10" + dm-1 · 10m-1 +...+ do · 10º it follows that: mod (n, 11) = mod (dm · 10m + dm-1 · 10m-1 + ..+ do · 10º, 11) = mod (dm · (-1)™ + dm-1 · (-1)"m-1 + ... = mod ((-1)"(dm – dm-1+ ·· [substitution] + do · (–1)º, 11) [mod(10, 11) = –1] + do · 1), 11) [factor out (-1)"] Therefore, mod (n, 11)=0 if and only if the alternating sums of the digits of

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 37E
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Please do Exercise 6.1.22 part abc and please show step by step

Proposition 6.1.19. An integer is divisible by 3 if and only if the sum of its digits is divisible by 3. Proposition 6.1.21. A number is divisible by 11 if and only if the alternat- ing sums of the digits is divisible by 11. (Note: alternating sums is where the signs of the number alternate when summing.) PROOF. Given an integer with digits do ... dn where the number is writeen as dndn-1...dıdo we can write n = dm · 10" + dm-1 · 10m-1 + ...+ do · 10º it follows that: mod (n, 11) = mod (dm · 10" + dm-1· 10m–1 +...+ do · 10°, 11) = mod(dm · (-1)™ + dm-1 · (–1)m-1+. = mod ((-1)" (dm – dm-1+ ·+do · 1), 11) [substitution] ..+ do · (–1)º, 11) [mod(10,11) = -1] [factor out (-1)"] ... Therefore, mod (n, 11)=0 if and only if the alternating sums of the digits of the number d, ... do is divisible by 11.
Transcribed Image Text:Proposition 6.1.19. An integer is divisible by 3 if and only if the sum of its digits is divisible by 3. Proposition 6.1.21. A number is divisible by 11 if and only if the alternat- ing sums of the digits is divisible by 11. (Note: alternating sums is where the signs of the number alternate when summing.) PROOF. Given an integer with digits do ... dn where the number is writeen as dndn-1...dıdo we can write n = dm · 10" + dm-1 · 10m-1 + ...+ do · 10º it follows that: mod (n, 11) = mod (dm · 10" + dm-1· 10m–1 +...+ do · 10°, 11) = mod(dm · (-1)™ + dm-1 · (–1)m-1+. = mod ((-1)" (dm – dm-1+ ·+do · 1), 11) [substitution] ..+ do · (–1)º, 11) [mod(10,11) = -1] [factor out (-1)"] ... Therefore, mod (n, 11)=0 if and only if the alternating sums of the digits of the number d, ... do is divisible by 11.
Exercise 6.1.22. (a) In Proposition 6.1.19 we showed that a number is divisible by 3 if and only if the sum of its digits is divisible by 3. Write a similar argument and state a proposition for a number that is divisible by 9. (b) Figure 6.1.1 shows a table giving the different powers of 10 mod base 37. Based on the results shown in Figure 6.1.1, propose a divisibility rule to check whether numbers are divisible by 37. Apply your rule to the following numbers: 17094, 411108, 365412 1 base 37 2 3 10*n(n>o Mod(An, 37) 1 10 10 6. 100 26 7 1000 1 10000 10 100000 26 10 1000000 1 11 10000000 12 100000000 10 26 13 Figure 6.1.1. Spreadsheet to compute the powers of 10 mod 37 (c) Create a spreadsheet similar to the the spreadsheet in Figure 6.1.1. Use your spreadsheet to find mod(10", 111) for 0 <n< 8. Come up with a proposition for numbers in base 111 and prove it similarly the divisibility rule for numbers in base 11 was proved in Proposition 6.1.21.
Transcribed Image Text:Exercise 6.1.22. (a) In Proposition 6.1.19 we showed that a number is divisible by 3 if and only if the sum of its digits is divisible by 3. Write a similar argument and state a proposition for a number that is divisible by 9. (b) Figure 6.1.1 shows a table giving the different powers of 10 mod base 37. Based on the results shown in Figure 6.1.1, propose a divisibility rule to check whether numbers are divisible by 37. Apply your rule to the following numbers: 17094, 411108, 365412 1 base 37 2 3 10*n(n>o Mod(An, 37) 1 10 10 6. 100 26 7 1000 1 10000 10 100000 26 10 1000000 1 11 10000000 12 100000000 10 26 13 Figure 6.1.1. Spreadsheet to compute the powers of 10 mod 37 (c) Create a spreadsheet similar to the the spreadsheet in Figure 6.1.1. Use your spreadsheet to find mod(10", 111) for 0 <n< 8. Come up with a proposition for numbers in base 111 and prove it similarly the divisibility rule for numbers in base 11 was proved in Proposition 6.1.21.
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