Problem 310
The following conjectures were made by students.
Which do you think are false? Give a counterexample for each conjecture if possible.
a. Only half of the nonzero even numbers up to 100 are divisible by 4.
b. A number is divisible by 8 if it is divisible by 4 and divisible by 2.
c. If 12 divides a number, 6 also divides the number.
d. If a number is divisible by 4 and 6, it must be divisible by 24.
Solution
a. True, because every second number is divisible by 4.
b. False, Consider 28. Both 2 and 4 are factors of 28 and thus divide 28 but 8 is not a factor of 28.
c. True,
d. False, Both 4 and 6 divide 36 but 24 is not a factor of 36.