Question

Use proof by exhaustion or disproof by counterexample to prove or disprove the following daims.

(a) For every integer n from 1 to 10 inclusive, the sum of the factors of n (excluding n itself), is
never greater than n.
(b) For every integer n from 10 to 20 inclusive, the sum of the factors of n (excluding n itself), is
never greater than n.

Answer 1

Claim (a) is proven to be true using proof by exhaustion, while claim (b) is disproven using a counterexample.

Step-by-step explanation:

In this question, we need to prove or disprove two claims using proof by exhaustion or disproof by counterexample. Let's examine each claim separately:

(a) For every integer n from 1 to 10 inclusive, the sum of the factors of n (excluding n itself) is never greater than n. To prove this claim, we can calculate the sum of factors for each integer from 1 to 10:

1. Factors of 1: None
2. Factors of 2: 1
3. Factors of 3: 1
4. Factors of 4: 1 + 2 = 3
5. Factors of 5: 1
6. Factors of 6: 1 + 2 + 3 = 6
7. Factors of 7: 1
8. Factors of 8: 1 + 2 + 4 = 7
9. Factors of 9: 1 + 3 = 4
10. Factors of 10: 1 + 2 + 5 = 8

As we can see, for every integer from 1 to 10, the sum of factors (excluding the number itself) is never greater than n. So the claim (a) is true.

(b) For every integer n from 10 to 20 inclusive, the sum of the factors of n (excluding n itself) is never greater than n. To prove this claim, we can calculate the sum of factors for each integer from 10 to 20:

1. Factors of 10: 1 + 2 + 5 = 8
2. Factors of 11: 1
3. Factors of 12: 1 + 2 + 3 + 4 + 6 = 16
4. Factors of 13: 1
5. Factors of 14: 1 + 2 + 7 = 10
6. Factors of 15: 1 + 3 + 5 = 9
7. Factors of 16: 1 + 2 + 4 + 8 = 15
8. Factors of 17: 1
9. Factors of 18: 1 + 2 + 3 + 6 + 9 = 21
10. Factors of 19: 1
11. Factors of 20: 1 + 2 + 4 + 5 + 10 = 22

As we can see, for some integers from 10 to 20, the sum of factors (excluding the number itself) is greater than n. So the claim (b) is false.