**CAT Questions on Factors of a Number:**

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**Question 1:** A “tragic number” is a number which can be expressed as the sum of three of its factors. For example, 6 can be expressed as the sum of 1, 2 and 3. How many tragic numbers are there that are less than 50?

a) 6

b) 7

c) 8

d) 9

**Question 2: **If N = 1980, Find the number and sum of its even factors.

a) 28, 6552

b) 24, 5616

c) 24, 4630

d) 28, 5672

**Question 3:** How many integers are both multiples of $125^{3124}$ and factors of $125^{3127}$?

a) 3

b) 10

c) 4

d) 11

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**Question 4:** What is the number of even factors of 36000 which are divisible by 9 but not by 36?

a) 20

b) 4

c) 10

d) 12

**Question 5: **How many factors of 36288 are perfect cubes?

a) 9

b) 4

c) 6

d) 8

Factors of a number – Formulas for CAT

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**Solutions for CAT Questions on Factors of a Number:**

**Solutions:**

**1) Answer (C)**

The factors of a number ‘x’ can be of the form $\frac{x}{2}$ , $\frac{x}{3}$ , $\frac{x}{5}$ , ….

If 2 is not a factor of the number then the highest three factors of the number can be $\frac{x}{3}$, $\frac{x}{5}$, $\frac{x}{7}$. The sum of these three is less than x. So 2 has to be a factor of the number.

If 3 is not a factor of the number the the highest three factors of the number can be $\frac{x}{2}$, $\frac{x}{4}$, $\frac{x}{5}$. The sum of these three is less than x. So 3 has to be a factor of the number.

If 2 and 3 are factors, 6 is also a factor.

Also the sum of $\frac{x}{2}$, $\frac{x}{3}$, $\frac{x}{6}$ is exactly equal to x.

So all numbers which are multiples of 6 are tragic numbers.

There are 8 such numbers which are below 50.

**2) Answer (B)**

1980 = $2^2 * 3^2 * 11 * 5$

Number of even factors = Total number of factors – Number of odd factors.

= (2+1)(2+1)(1+1)(1+1) – (2+1)(1+1)(1+1) = 24

Sum of even factors = Sum of all the factors – sum of odd factors

= $(\frac {2^{(2+1)} -1} {(2-1)})$ x $(\frac {3^{(2+1)} -1} {(3-1)})$ x $(\frac {11^{(1+1)} -1} {(11-1)})$ x $(\frac {5^{(1+1)} -1} {(5-1)})$ – $(\frac {3^{(2+1)} -1} {(3-1)})$ x $(\frac {11^{(1+1)} -1} {(11-1)})$ x $(\frac {5^{(1+1)} -1} {(5-1)})$

= 7*13*12*6 – 13*12*6 = 5616

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**3) Answer (B)**

$125^{3124}$ = $5^{9372}$

$125^{3127}$ = $5^{9381}$

Thus, the factors of $5^{9381}$ which are multiples of $5^{9372}$ are $5^{9372}$,$5^{9373}$ … $5^{9381}$. Thus, there are 10 such integers

**4) Answer (B)**

$36000 = 2^{5}*3^{2}*5^{3}$

Since we are talking of even factors, there must be at least one 2 in the required factors.

Since the number is divisible by 9, we must have both the threes.

We cannot have more than 1 two as it will make the number divisible by 36.

So we have 1 way of choosing 2, 1 way of choosing 3, 4 ways of choosing 5.

Thus the required number of factors are

1*1*4 = 4

**5) Answer (C)**

$36288 = 2^6 * 3^4 * 7$

For any perfect cube, all the powers of its prime numbers have to be multiples of 3.

So, if the factor is of form $2^a * 3^b * 7^c$, a can take values 0, 3, 6

And b can take values 0, 3

And c can take value 0.

==> There are 3*2*1 = 6 possibilities.