numbers

show that the square of any prime number reduced by one is always divisible by 24.




Answer:

Any prime number can be represented in the form 6x + 1 or 6x - 1.

So, if the number is 6x + 1

Then, (6x + 1)² = 36x² + 12x + 1

So, (6x + 1)² – 1 = 36x² + 12x = 12x (3x + 1)

If x is even then 3x + 1 is odd or visa versa. So, x (3x + 1) is always a factor of 2.

Therefore, (6x + 1)² – 1 = 24 k, where 2k =  x (3x + 1)

Similarly, if the number is 6x - 1

Then, (6x - 1)² = 36x² - 12x + 1

So, (6x - 1)² – 1 = 36x² - 12x = 12x (3x - 1)

If x is even then 3x - 1 is odd or visa versa. So, x (3x - 1) is always a factor of 2.

Therefore, (6x - 1)² – 1 = 24 p, where 2p =  x (3x - 1)

So, the square of a prime number reduced by one is always divisible by 24.