This question asks you to prove facts about numbers divisible by 9. Do NOT use proof by induction. (a) Prove that 10" – 1 is divisible by 9. Hint: see the challenge problem from the week 9 worksheet. (b) The decimal representation of a number is the representation that we are used to, where the i'h digit represents a value times 10°. For example, the number 321567 = 3 · 10° + 2.104 +1.103 + 5. 102 + 6 · 10' + 7· 10°. We will denote the digit in the i" position as di. In this example, do = 7, di = 6, d2 = 5, dz = 1, d4 = 2, d5 = 3. Given a number x in decimal representation, let s represent the sum of the digits of x. Prove that x – s is divisible by 9.

Question
This question asks you to prove facts about numbers divisible by 9. Do NOT use proof by induction.
(a) Prove that 10" – 1 is divisible by 9. Hint: see the challenge problem from the week 9 worksheet.
(b) The decimal representation of a number is the representation that we are used to, where the i'h digit represents a
value times 10°. For example, the number 321567 = 3 · 10° + 2.104 +1.103 + 5. 102 + 6 · 10' + 7· 10°. We will
denote the digit in the i" position as di. In this example, do = 7, di = 6, d2 = 5, dz = 1, d4 = 2, d5 = 3.
Given a number x in decimal representation, let s represent the sum of the digits of x. Prove that x – s is divisible
by 9.
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Transcribed Image Text

This question asks you to prove facts about numbers divisible by 9. Do NOT use proof by induction. (a) Prove that 10" – 1 is divisible by 9. Hint: see the challenge problem from the week 9 worksheet. (b) The decimal representation of a number is the representation that we are used to, where the i'h digit represents a value times 10°. For example, the number 321567 = 3 · 10° + 2.104 +1.103 + 5. 102 + 6 · 10' + 7· 10°. We will denote the digit in the i" position as di. In this example, do = 7, di = 6, d2 = 5, dz = 1, d4 = 2, d5 = 3. Given a number x in decimal representation, let s represent the sum of the digits of x. Prove that x – s is divisible by 9.

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