JEE Novice - (8) (changed)

Number Theory Level 2

$\Large N=7^{p+4}\times 5^q \times 2^3$

If $N$ is a perfect cube where $p,q \in \mathbb{N}$ , find the smallest possible value of $p + q$ .

This question is a part of JEE Novices .

The answer is 5.

3 solutions

Trevor Arashiro
May 26, 2015

each number is a prime, so it must be expressed as $n^a$ where a is an integer. If it's not, we won't have a perfect cube. Each number's power must be a multiple of 3, so that N will be a perfect cube.

Thus $3|p+4\Rightarrow p=2$

$3|q\Rightarrow q=3$

Abc Xyz
Feb 26, 2016

Here p and q are natural numbers. Therefore $p,q \geq 0$ .

Also we need minimum values. So q=3........since q can't be 0 and N is a perfect cube.

But in the case of p, if p+4=3 then p=-1 which is not possible.So we take the next multiple of 3 which is 6.

p+4=6. Therefore p=2

So the minimum value for p + q =(2+3) = $\boxed{5}$

Manish Dash
May 26, 2015

Since N is a perfect cube hence N must be a product of cubes of its prime factors. Moreover, since p and q are natural numbers, therefore p>0 and q>0.

We know 7, 5 and 2 are primes hence their exponents must be multiples of 3
Therefore, Minimum value of p = 2 and that of q =3 Hence p+q = 5

Moderator note:

Why? Why can't $q$ be smaller than 3? Why can't $p$ be smaller than 2?

Reply to the challenge master: I have changed my solution accordingly. I hope it is ok now.

Manish Dash - 6 years ago

p and q can be less than 2 and 3 respectively. They can easily attain negative values but it is mentioned in the question that they are natural numbers. Thus, they cannot be less than their least positive values , which are 2 and 3.

Abhijeet Verma - 6 years ago

JEE Novice - (8) (changed)

This question is a part of JEE Novices .

The answer is 5.

3 solutions

Moderator note:

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