Math 11
More on deductive proofs – divisibility and number patterns
1. Set up your on-line text book (look at the inside cover of your hard copy)
2. Turn (or scroll) to section 1.4, page 30. Review the deductive proof of “For a two digit number, if the sum of the digits is divisible by 3, then the original number is divisible by three”
3. Use the same logic to prove that, for a three digit number ABC, if the sum of the digits A+B+C is divisible by three, then the original number is as well.
4. What is the quick trick to check if a number is divisible by 5? Prove, for two digit numbers, this divisibility trick.
5. Play this little game:
• Choose any number.
• Multiply by 4.
• Add 10.
• Divide by 2.
• Subtract 5.
• Divide by 2.
• Add 3. a) Show inductively, using three examples, that the result is always 3 more than the chosen number. b) Prove deductively that the result is always 3 more than the chosen number.
6. Consider any odd integer. a) When it is squared, and then divided by 4, what is the result? E.g. 3 2 = 9, and 9/4 = ….? b) Repeat the above with a few different odd integers. c) Make a conjecture about the nature of squaring an odd integer and then dividing by 4. d) prove your conjecture
7. Make a conjecture about the product of two consecutive natural numbers, and then prove your conjecture.