which of the following is not a perfect square: 100856 and 945729 ? Explain the easiest and fastest way of checking it.

Find out their factors and if all the unique factors have even powers then it is a perfect square.

Below is the easy and fastest way to check the perfect square-

Squares of all integers are known as perfect squares. All perfect squares end in 1, 4, 5, 6, 9 or 00 (i.e. Even number of zeros). Therefore, a number that ends in 2, 3, 7 or 8 is not a perfect square.

For all the numbers ending in 1, 4, 5, 6, & 9 and for numbers ending in even zeros, then remove the zeros at the end of the number and apply following tests:

• Digital roots are 1, 4, 7 or 9. No number can be a perfect square unless its digital root is 1, 4, 7, or 9. You might already be familiar with computing digital roots. (To find digital root of a number, add all its digits. If this sum is more than 9, add the digits of this sum. The single digit obtained at the end is the digital root of the number.)
• If unit digit ends in 5, ten’s digit is always 2.
• If it ends in 6, ten’s digit is always odd (1, 3, 5, 7, and 9) otherwise it is always even. That is if it ends in 1, 4, and 9 the ten’s digit is always even (2, 4, 6, 8, 0).
• If a number is divisible by 4, its square leaves a remainder 0 when divided by 8.
• Square of even number not divisible by 4 leaves remainder 4 while square of an odd number always leaves remainder 1 when divided by 8.
• Total numbers of prime factors of a perfect square are always odd.