Euler's criterion
Alright kiddo, let me tell you a story about a guy called Euler and a rule he made about numbers. Euler was a really smart guy who liked to solve puzzles with numbers, and he made a rule called the Euler's criterion.
The rule helps us Figure out if a number is a square (like 4 or 9) or not. This is important because we can use this rule to solve other math problems too!
Here's how it works:
We start with a number, let's say "a", and a special number called "p". Now, we can find out if "a" is a square modulo "p" using Euler's criterion.
First, we find a number called "b". This "b" is the remainder of when we divide "a" by "p". So if "a" is 15 and "p" is 7, then "b" would be 1, because when 15 is divided by 7, the remainder is 1.
Next, we find a new number called "c". This "c" is equal to (p-1)/2. So if "p" is 7, then "c" would be 3 because (7-1)/2 is equal to 3.
Then, we raise "b" to the power of "c" and take the remainder when divided by "p". If the remainder is 1, then "a" is a square modulo "p". If the remainder is "-1", then "a" is not a square modulo "p".
Now, I know that might sound a bit complicated, but that's basically what Euler's criterion is! It's a rule that helps us figure out if a number is a square or not by using some special numbers and raising a number to a power.
Isn't math cool, kiddo?
The rule helps us Figure out if a number is a square (like 4 or 9) or not. This is important because we can use this rule to solve other math problems too!
Here's how it works:
We start with a number, let's say "a", and a special number called "p". Now, we can find out if "a" is a square modulo "p" using Euler's criterion.
First, we find a number called "b". This "b" is the remainder of when we divide "a" by "p". So if "a" is 15 and "p" is 7, then "b" would be 1, because when 15 is divided by 7, the remainder is 1.
Next, we find a new number called "c". This "c" is equal to (p-1)/2. So if "p" is 7, then "c" would be 3 because (7-1)/2 is equal to 3.
Then, we raise "b" to the power of "c" and take the remainder when divided by "p". If the remainder is 1, then "a" is a square modulo "p". If the remainder is "-1", then "a" is not a square modulo "p".
Now, I know that might sound a bit complicated, but that's basically what Euler's criterion is! It's a rule that helps us figure out if a number is a square or not by using some special numbers and raising a number to a power.
Isn't math cool, kiddo?