Question

find the number of factors of 172800 which are divisible by 12 but not by either 36 or 25

Answer 1

Explanation:

Given that:

What is the factor 172800 which are divisible by 12 but not by either 36 or 25

To find: Number of factors that are divisible by 12 but not by either 36 or 25

Solution:

To find Number of factors that are divisible by 12 but not by either 36 or 25

We have to do prime factorisation first

$172800 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times2 \times 3 \times 3 \times 3 \times 5 \times \\ \\ \bold{172800 = {2}^{7} \times {3}^{3} \times {5}^{2}}$

Divisibility rule:

• Divisible by 12: A number is divisible by 12,if it is divisible by both 3 and 4
• Divisible by 36: A number is divisible by 36,if it is divisible by both 4 and 9.
• Divisible by 25: A number is divisible by 25,if it is divisible by 5

To find factors which are divisible by 12 but not by 36 and 25

We have to take the factors which includes 2^n and 3 only,because if a number is divisible by 4,then it is surely divisible by 2

But don't take the numbers which have factors 3²,3³,5 and 5² because they includes the numbers which are divisible by 36 and 25.

So,

Take factors from

${2}^{7} \times {3}^{3}$

Numbers formed are like ,which are divisible by 12 but not by 36 and 25

${2}^{2} \times {3}^{1} \\ {2}^{3} \times {3}^{1} \\ {2}^{4} \times {3}^{1} \\ {2}^{5} \times {3}^{1}\\{2}^{6} \times {3}^{1}\\ {2}^{7} \times {3}^{1}$

Thus,

6 numbers(12,24,48,96,192,384) are there in the factors of 172800 which are divisible by 12 but not divisible by 36 and 25.

Hope it helps you.