The squareroot of 4356 is an integer. WITHOUT a calculator I was able to elimina

Question

The squareroot of 4356 is an integer. WITHOUT a calculator I was able to eliminate some possibilities. I knew it had to be even. Why? I knew it was greater than 30, oh wait 40, no actually 50, really it hit me that it is greater than 60 and less than 70. Why? Wow that only leaves four choices. 62, 64, 66, and 68. I know two of these are not it right away. Why and which ones? Hmmm.....That means the right answer is? Now, try the process with 8464. Explain how you eliminated the possibilities each time.

Explanation / Answer

Solution:

to the square root of 4356.

we can do this by trial and error.

we try some simple values to bracket the possibilities:
60*60 = 3600 (too small)
70*70 = 4900 (too big)

Now, 4356 is an even number.

So you know that the square root cannot be an odd number. Because the product of two odd numbers is always another odd number

So, the answer can only be 62, 64, 66, or 68.

Now, since 4356 ends in a last digit '6', the only one of those four possibilities that can end in a '6' is 66 * 66.

So, 66 is the answer.

Similerly for 8464

to the square root of 8464.

Try some simple values to bracket the possibilities:
90 *90 = 8100 (too small)
96*96 = 9216 (too big)

Now, 8464 is an even number.

So we know that the square root cannot be an odd number. Because the product of two odd numbers is always another odd number
So, the answer can only be 92, or 94, .

Now, since 8464 ends in a last digit with '4', the only one of those four possibilities that can end in a '4' is 92 * 92. So, 92 is the answer

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