Question

A number ‘n’ is written as p^2q^4r^6 , where p q r , & are prime factors. Find the square root of this number?

Answer 1

Answer:

225

It is given that LCM(p,q)=r

2

t

4

s

2

.

That is, at least one of p and q must have r

2

,t

4

and s

2

in their prime factorizations.

Now, consider the cases for power of r as follows:

Case 1: p contains r

2

then q has r

k

with k=(0,1).

That is, number of ways=2.

Case 2: q contains r

2

then p has r

k

with k=(0,1).

That is, number of ways=2.

Case 3: Both p and q contains r

2

Then, number of ways=1.

Therefore, exponent of r may be chosen in 2+2+1=5 ways.

Similarly, exponent of t may be chosen in 4+4+1=9 ways and exponent of s may be chosen in 2+2+1=5 ways

Thus, the total number of ways is:

5×9×5=225

Hence, the number of the ordered pair (p,q) is 225.