IPMAT Question Paper 2021 | IPM Indore Quantitative Ability

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Question 1 : The number of positive integers that divide (1890).(130).(170) and are not divisible by 45 is ________

Video Explanation

Explanatory Answer

We start with prime factorization of the number
So, (1890).(130).(170) can be rewritten as
2³. 3³. 5³. 7. 13. 17
And so any factor of this number has to be of the form 2^a. 3^b. 5^c. 7^d. 13^e. 17^f
Where a, b, and c can be any number among 0, 1, 2, 3
And d, e, and f can be any among 0 or 1.
From here we know that there are 4 possibilities each for a, b and c, and for c, d, and e we have 2 possibilities.
So there are 4.4.4.2.2.2 factors
= 512 factors
From all the factors of the number, we have to remove the number of multiples of 45.
45 = 3² . 5
So for the number to be a multiple of 45, it should have a factor of 3² and a 5 in it.
So the powers can be
a can be 0, 1, 2, 3 = 4 values
b can be 2, 3 = 2 values
c can be 1, 2, 3 = 3 values
d can be 0, 1 = 2 values
e can be 0, 1 = 2 values
f can be 0, 1 = 2 values
4.2.3.2.2.2 = 192 numbers are multiples of 45
So among all the 512 factors, if we remove all the factors of 45 we will get the total factors that are not divisible by 45.

So, 512 - 192 = 320 factors of the given number that are not divisible by 45

The question is " The number of positive integers that divide (1890).(130).(170) and are not divisible by 45 is ________ "

Hence, the answer is '320'

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