11 Questions Involving Divisibility, Functions, and Modular Arithmetic

6. Let f: M ->N be a function from M to N, and any different elements of M have different images in B. Is f onto or one-to-one function?

7. What is the inverse of the following function?
f = {(1, x), (2, y), (3, z), (4, t)}

8. Let f ={(1,2),(2,3),(3,4),(4,1)} and g={(1,3),(2,1),(3,4),(4,2)}
What is the composite function g ° f?

2. The equation 15 = (2 x 4) + 7 is correct.

3. The equation -15 = ((-4) x 4) + 1 with a = 15, b = -4, q = 4, and r = 1 satisfies the requirements of the Division Algorithm.

4. If a, b > 0 and q, r are chosen to satisfy the requirements of the Division Algorithm, then q = ┌a/b┐?

5. Given two consecutive integers a, a + 1, one of them must be divisible by 2?

6. If a, b, c are nonzero integers such that c | a and c | b, then gcd(a, b) ≤ c?


8. If gcd(a, b) = lcm(a, b), then a = b?

9. If a | c and b | c, then lcm(a, b) ≤ c?

10. 55≡7 (mod 3)?

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5. Given two consecutive integers a, a + 1, one of them must be divisible by 2?
5. Yes. There are two cases:
a. a is even and then is divisible by 2.
b. a is odd, but this case a+1 is even, thus divisible by 2.

6. If a, b, c are nonzero integers such that c | a and c | b, then gcd(a, b) ≤ c?
6.
It is false, because we can take the following counterexample:
a=15, b=45 and c=3.
We have that 3/15 and 3/45. Also, GCD(15,45)=15 which is greater than 3.

8. If gcd(a, b) = lcm(a, b), then a = b?
8. The conclusion is true only for positive numbers a and b....