# Find a whole number that is both perfect square and cube found between 2000 and 10000

#### Sparticusretiarius

##### New member
"Find a whole number that is both perfect square and cube found between 2000 and 10000?"

I am kind of stumped on this question. If anyone can help me that would great.

#### Subhotosh Khan

##### Super Moderator
Staff member
"Find a whole number that is both perfect square and cube found between 2000 and 10000?"

I am kind of stumped on this question. If anyone can help me that would great.
Brute Force!!

How many integer square numbers are there between 2000 ↔ 10000 (452 ↔ 1002) - list those

How many integer cubic numbers are there between 2000 ↔ 10000 (133 ↔ 213) - list those

Hint: How many "square numbers" do you have within 13 ↔ 21?

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#### lookagain

##### Senior Member
"Find a whole number that is both perfect square and cube found between 2000 and 10000?"

"Brute force" was stated, but it is not required.

Because it is a perfect square, it is of the form $$\displaystyle \ N^2$$.

Because it is also a perfect cube, it is also of the form $$\displaystyle \ M^3$$.

$$\displaystyle x^6 \ = \ (x^3)^2 \ = \ (x^2)^3 \$$ satisfies both. (The exponent on x is the least common multiple of the other two exponents.)

Then we have:

$$\displaystyle 2,000 \ < x^6 \ < 10,000$$

$$\displaystyle \sqrt[6]{2,000} \ < x \ < \sqrt[6]{10,000}$$

$$\displaystyle 3.5... \ < x \ < 4.6...$$

What can you conclude?

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#### JeffM

##### Elite Member
It is obvious that

$$\displaystyle a \in \mathbb Z \implies a^2,\ a^3, a^6 \in \mathbb Z \text { and } (a^2)^3 \equiv a^6 \equiv (a^3)^2.$$

It is not obvious, however, that

$$\displaystyle b^2 = c^3 \in \mathbb Z \implies \sqrt{b} \in \mathbb Z.$$

It may be true, but it requires a proof.