Vijayeta wrote:

225, or 15^2, is the first perfect square that begins with two 2s. What is the sum of the digits of the next perfect square to begin with two 2s?

A. 9

B. 12

C. 13

D. 14

E. 17

I used a slightly unconventional approach.

First we need to eliminate options B,D and E immediately. (I will explain why at the end of the solution).

We will be left with option A and C.

Let's take option A.

If the first two digits are 22, then the sum of digits of the first two digits is already 4 and we are left with (9-4) = 5 for the last two digits.

The possible cases would be 2205, 2250, 2214, 2241, 2223 and 2232

Again t

hese can be discarded easily because number ending with 5, will have 2 as the tens digits. Number ending with 0 must have 0 as the tens digit. Number ending with 4 number have an even number as the tens digit. Number ending with 3 and 2 cannot be perfect square. A slight check might be needed for 2241, but then this can also be easily eliminated because 40^2 is 1600 and 41^2 cannot be 2241.

Since none of the numbers can be squares and have sum of digits as 9, thus

the answer has to be Option C.

Note: I am not trying to find the actual number. All I am doing is using the properties of a perfect square to eliminate the options.

Now if you are wondering why I eliminated options B, D and E. then here is the reason.

Take a few perfect squares and try to find the digital sum of the number. By digital sum, I mean, keep on adding the digits, till you get a single number.

1^2 = 1 ( Sum is 1)

2^2 = 4 ( Sum is 4)

3^2 = 9 ( Sum is 9)

4^2 = 16 ( Sum is 7)

5^2 = 25 ( Sum is 7)

6^2 = 36 ( Sum is 9)

10^2 = 100 ( Sum is 1)

11^2 = 121 ( Sum is 4)

13^2 = 196 ( Sum is 16 = 1+ 6 = 7)

.....

You can keep on trying with as many perfect squares as possible, there will be one pattern that you will notice -

The digital sum is either 1 or 4 or 7 or 9 and nothing else. You will not get any other number.

I used to this property to eliminate the options.

The digital sum of all the 5 options are -

A. 9 (digital sum is 9, this could be the answer)

B. 12 ( digital sum is 3, this cannot be the answer)

C. 13 ( digital sum is 4, this can be the answer)

D. 14 ( digital sum is 5, this cannot be the answer)

E. 17 ( digital sum is 8, this cannot be the answer)

I was kind of unlucky and had to do some calculation to eliminate 9. If 9 would not have been in the answer options, I would have straight away marked 13 as the answer, without thinking twice. :D

Regards,

Saquib

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