Math 11
More on inductive and deductive reasoning and conjecture
1. Consider multiplying a two digit number by 11. Here are a few specific examples:
17(11) = 187
23(11) = 253
41(11) = 451
Sept 5
62(11) = 682
a) A conjecture states that when multiplying a two digit number by 11, the first and last digits of the product are the digits of the
original number, e.g. 62(11) = 682. Is this conjecture reasonable?
b) can you find a counterexample that disproves this example?
c) if this conjecture seems to be valid, can you prove it?
d) if this conjecture is invalid, can you state the conditions under which it is not?
2. Do you agree or disagree with each conjecture? If you disagree, find a counterexample
a) the sum of a multiple of 3 and a multiple of 6 must be a multiple of 6
b) in any multiple of 9, the sum of the digits must also be 9
c) all natural numbers can be written as the sum of consecutive natural numbers, e.g. 9 = 4 + 5 , 12 = 3 + 4 + 5
3. Conjectures don’t always have to be about math, you know.
a) This map has no roads, urban areas, or human features on it at all. However, the map is
of an area that contains a town of 10,000 people. Make a conjecture about where this town
might be, and justify it.
b) In 1989, the image at right became
one of the most famous photos ever taken. Make a conjecture about the
events happening at the time it was shot.
4. Evens and Primes: Goldblach’s conjecture says that every even number greater than 2 can be written as the sum of two prime
numbers. Research on the internets…. Is Goldblach’s conjecture true?
5. Conjecture: as long as n is a positive integer, the expression n2 + n + 2 will never yield an odd number. Do you agree or disagree?
Justify your answer. This doesn’t mean “prove” – just justify
6. Deductive reasoning. Remember, deductive reasoning is about valid, sensible assumptions, making logical arguments, and
arriving and specific conclusions. Use some deductive reasoning for each:
a) All cats are mammals. Mammals are vertebrates. Rex is a cat. Make a conclusion about Rex
b) Yoga pants are comfortable. Woods is wearing comfortable pants. Woods must be wearing Yoga pants. Valid?
c) Lifting weight strengthens muscles. Muscles make you strong. Strength improves balance. John lifts weights. Make a
conclusion about John.
7. Deductive reasoning and mathematical proof: A conjecture states that the sum of 5 consecutive integers is equal to 5 times the
average (mean) of these integers. Prove this conjecture using the general, deductive proof techniques that we have seen.
8. One last kick at the can. Prove that the sum of three consecutive integers is divisible by 3.