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# [Smart Math] Arithmetic Problem 14

Here’s and example of a SMART MATH problem for ARITHMETIC.

### Problem

What is the least number which must be subtracted from 10420 to make it a perfect square?

1. 219
2. 200
3. 189
4. 25
5. 16

### The Usual Method

10420 can be written as:

10420 = 10000 + 420

The square root of 10000 = 100

Now, by trail and check method, check for squares of numbers greater than 100.

$120^{2}=14400$ as 14400 > 10420 hence the square root should be < 120.

$110^{2}=12100$ as 12100 > 10420 hence the square root should be < 110.

$105^{2}=11025$ as 11025 > 10420 hence the square root should be < 105.

$102^{2}=10404$ as 10404 < 10420 hence the required square root should be 102.

To bring 10420 to 10404, we have to subtract 10420 – 10404 = 16.

(Ans: 5)

Estimated Time to arrive at the answer = 60 seconds.

### Using Technique

Simply subtract each option from 10420 starting with the least number and check for perfect square as follows:

e. 10420 – 16 = 10404 = $100^{2}+2\times 100\times 2+2^{2}=102^{2}$ (Perfect Square)

(Note: We are taking option ‘5’ first as it is the one with the lowest number.)

No need to check any further.

Hence, option ‘5’.

(Ans: 5)

Estimated Time to arrive at the answer = 10 seconds.
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