# Number Theory. Argue Is not the square of an integer.

• celtics777
In summary, the conversation discusses arguing that (17^4)*(5^10)*(3^5) is not the square of an integer. The solution proposed involves breaking down the expression and showing that each factor is not a square. However, this is not an efficient explanation as it does not consider the fact that 3, 5, and 17 are prime numbers. The importance of this is highlighted by considering a similar question for \sqrt{(8)(18)}.

## Homework Statement

Argue that (17^4)*(5^10)*(3^5) is not the square of an integer.

N/A?

## The Attempt at a Solution

Do I break these up, and show that each is not a square? I'm not sure if that would be correct, but sqrt(17^4)=289 * sqrt(5^10)=3125 * sqrt(243)=15.5884 =...

Since sqrt(243)=15.5884 and is not an integer then the above is not the square of an integer. Is this an efficient explanation?

Well, it certainly doesn't show any understanding of the problem! Look at the exponents: $\sqrt{17^4}= (17^4)^{1/2}= 17^2$. $\sqrt{5^{10}}= (5^{10})^{1/2}= 5^5$. What about $\sqrt{3^5}$?

Do you see why the fact that 3, 5, and 17 are prime numbers is important?
(Consider the same question about $\sqrt{(8)(18)}$.)

## 1. What is Number Theory?

Number Theory is a branch of mathematics that deals with the properties and relationships of numbers. It focuses on studying the patterns and structures of numbers and their interactions with each other.

## 2. How is Number Theory different from other branches of mathematics?

Number Theory is unique in that it primarily focuses on the properties and relationships of whole numbers, rather than using symbols and equations to represent abstract concepts. It also has applications in various fields, such as cryptography and computer science.

## 3. What does it mean for a number to be the "square" of another number?

A number is considered the square of another number if it can be expressed as the product of that number multiplied by itself. For example, 9 is the square of 3 since 9 = 3 x 3.

## 4. Is it possible for the square of an integer to be a non-integer number?

No, the square of any integer will always result in an integer. This is because when we multiply two integers, the product will always be an integer.

## 5. Can you prove that the square of an integer is always an integer?

Yes, we can prove this using the definition of a square number. Let's say we have an integer, n. Its square would be n x n. Since n is an integer, it can be expressed as a fraction of two integers, p and q (n = p/q). Therefore, the square of n can be written as (p/q) x (p/q) = p^2/q^2. Since p and q are also integers, their square will result in an integer. Therefore, the square of an integer will always be an integer.