For each statement below, determine whether it is always true, sometimes true, or never true. Explain your answer. Class 8

For each statement below, determine whether it is always true, sometimes true, or never true. Explain your answer. Class 8

Question 1.

For each statement below, determine whether it is always true, sometimes true, or never true. Explain your answer. Mention examples and non-examples as appropriate. Justify your claim using algebra. Class 8

(i) The sum of two even numbers is a multiple of 3.

(ii) If a number is not divisible by 18, then it is also not divisible by 9.

(iii) If two numbers are not divisible by 6, then their sum is not divisible by 6.

(iv) The sum of a multiple of 6 and a multiple of 9 is a multiple of 3.

(v) The sum of a multiple of 6 and a multiple of 3 is a multiple of 9.

Solution:

(i) Sometimes true. For example, 4 + 8 = 12 is divisible by 3, but 6 + 8 = 14 is not divisible by 3.

Algebra : Let 2m and 2n be two even numbers. Sum = 2(m + n). This will be divisible by 3, if m + n is divisible by 3, which is true, when m + n = 3, 6, 9 and not true when m + n = 4, 5, 7, 10, 11, ...

(ii) Sometimes true. For example, 27 is not divisible by 18, but it is divisible by 9. However, 35 is not divisible by both 18 and 19.

Algebra : The number 9m is divisible by 9 for m = 1, 2, 3, ... but 9m is not divisible by 18 for m = 1, 3, 5, ...

(iii) Sometimes true. For example, 5 and 9 are not divisible by 6 and also 5 + 9 = 14 is also not divisible by 6. But 9 and 21 are not divisible by 6, but 9 + 21 = 30 is divisible by 6.

Algebra : Let x and y be not divisible by 6. Then, x + y may be a multiple of 6 for some values x and y and it may not be a multiple of 6 for some values of x and y.

(iv) Always true.

Algebra : Let the numbers be 6a and 96. So, 6a + 96 = 3 (2a + 36), which will always be divisible by 3.

(v) Sometimes true. For example, 18 is a multiple of both 6 and 3. It is also divisible by 9. But 12 is a multiple of both 6 and 3, but not divisible by 9.

Algebra : A number divisible by 6 and 3 will be of the form 6x and it will be multiple of 9 for x = 3, 6, 9, ...