State whether the following statements are true (T) or false (F).
Factors of even numbers must be even.
False. Even numbers can have odd factors. For example, 6 is even but has factors 1, 2, 3, 6, where 1 and 3 are odd.
Multiples of odd numbers cannot be even.
False. Multiples of odd numbers can be even. For example, =6 is an even multiple of the odd number 3.
Factors of odd numbers cannot be even.
True. Factors of odd numbers are always odd. If an odd number had an even factor, dividing by that even factor would not give a whole number.
One of the common multiples of two consecutive numbers is their product.
True. The product of two consecutive numbers is always a common multiple of both. For example, =12 is a common multiple of 3 and 4.
The only common factor of any two consecutive numbers is 1.
True. Two consecutive numbers always have only 1 as their common factor. For example, 8 and 9 share only 1 as a common factor.
0 cannot be a factor of any number.
True. 0 cannot be a factor of any number because division by 0 is undefined. A factor must divide the number evenly, but dividing by 0 is not possible.
Explanation
(a) Even numbers like 6 have factors 1, 2, 3, 6. The factors 1 and 3 are odd, proving that factors of even numbers need not all be even. (b) When an odd number is multiplied by an even number, the result is even. For instance, =12, =30 are even multiples of odd numbers. (c) Odd numbers cannot have even factors because dividing an odd number by an even number never gives a whole number. All factors of odd numbers are odd. (d) When we multiply two consecutive numbers like 3 and 4, we get 12. This is divisible by both 3 (12÷3=4) and 4 (12÷4=3), making it a common multiple. (e) Consecutive numbers like 8 and 9 have no common factors other than 1. This is because consecutive numbers differ by 1, so they cannot share any factor greater than 1. (f) Division by zero is mathematically undefined. A factor must divide a number to give a whole number quotient, but dividing any number by 0 is not defined, so 0 cannot be a factor.