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# [Smart Math] Algebra Problem 13

Here’s and example of a SMART MATH problem for ALGEBRA. ### Problem

What is the number whose square is equal to the difference of the squares of 1884 and 1020?

1. 29.4
2. 538.9
3. 8165
4. 1584
5. 746496

### The Usual Method

First find the difference of squares of 1884 and 1020: $1884^{2}-1020^{2}$ = (1884 – 1020) (1884 + 1020)

= 864 x 2904

= 2509056

Now, taking the square root of 2,509,056:

Factorize 2509056, we get:

2509056 = $2^{8}\times 3^{4}\times 11^{2}$ $\therefore \sqrt{\text{2509056}}=\sqrt{2^{8}\times 3^{4}\times 11^{2}}=2^{4}\times 3^{2}\times 11=16\times 9\times 11=144\times 11=1584$

(Ans: 4)

Estimated Time to arrive at the answer = 150 seconds.

### Using Technique

The last digit of the square of 1884 will be 6 ( $\because 4^{2}$ = 16, Last digit = 6)

Similarly, the last digit of the square of 1020 will be 0.

Hence, last digit of difference between the squares of the numbers = 6 – 0 = 6

This eliminates options ‘2’ and ‘3’ as both when squares will have its last digit not equal to 6.

From amongst the remaining options, the option ‘4’ is most likely to be the answer as the number whose square is equal to the difference of squares of 1884 and 1020, will be also a 4 digit number lying between 1884 and 1020. The only 4 digit number satisfying the conditions is 1584 i.e. option ‘4’.

(Ans: 4)

Estimated Time to arrive at the answer = 15 seconds.
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