# Square of Numbers:

#### Overview:

 Topic Included: Formulas, Definitions & Exmaples. Main Topic: Quantitative Aptitude. Quantitative Aptitude Sub-topic: Simplification Aptitude Notes & Questions. Questions for practice: 10 Questions & Answers with Solutions.

Square of numbers always help during the solving the questions of simplification, so try to remember the square of numbers at least till 50. If you remember the squares, it will help you to speed up your solving capacity. If you can't remember then use the method given below.

 $$1^2$$ 1 $$2^2$$ 4 $$3^2$$ 9 $$4^2$$ 16 $$5^2$$ 25 $$6^2$$ 36 $$7^2$$ 49 $$8^2$$ 64 $$9^2$$ 81 $$10^2$$ 100 $$11^2$$ 121 $$12^2$$ 144 $$13^2$$ 169 $$14^2$$ 196 $$15^2$$ 225 $$16^2$$ 256 $$17^2$$ 289 $$18^2$$ 324 $$19^2$$ 361 $$20^2$$ 400 $$21^2$$ 441 $$22^2$$ 484 $$23^2$$ 529 $$24^2$$ 576 $$25^2$$ 625
 $$26^2$$ 676 $$27^2$$ 729 $$28^2$$ 784 $$29^2$$ 841 $$30^2$$ 900 $$31^2$$ 961 $$32^2$$ 1024 $$33^2$$ 1089 $$34^2$$ 1156 $$35^2$$ 1225 $$36^2$$ 1296 $$37^2$$ 1369 $$38^2$$ 1444 $$39^2$$ 1521 $$40^2$$ 1600 $$41^2$$ 1681 $$42^2$$ 1764 $$43^2$$ 1849 $$44^2$$ 1936 $$45^2$$ 2025 $$46^2$$ 2116 $$47^2$$ 2209 $$48^2$$ 2304 $$49^2$$ 2401 $$50^2$$ 2500

Square of Numbers between 10 to 19:

Example(1): Find the square of 12?

Solution: $$12 = 12 + 2 \ | \ 2^2$$ $$= 14 \ | \ 4$$ $$= 144$$

Example(2): Find the square of 14?

Solution: $$14 = 14 + 4 \ | \ 4^2$$ $$= 18 \ | \ 16$$ $$= 18 + 1 \ | \ 6$$ $$= 19 \ | \ 6$$ $$= 196$$

Example(3): Find the square of 15?

Solution: $$15 = 15 + 5 \ | \ 5^2$$ $$= 20 \ | \ 25$$ $$= 20 + 2 \ | \ 5$$ $$= 22 \ | \ 5$$ $$= 225$$

Square of Numbers between 20 to 29:

Example(1): Find the square of 20?

Solution: $$20 = 20 + 0 \ | \ 0^2$$ $$= 20 \times 2 \ | \ 0$$ $$= 40 \ | \ 0$$ $$= 400$$

Example(2): Find the square of 22?

Solution: $$22 = 22 + 2 \ | \ 2^2$$ $$= 24 \times 2 \ | \ 4$$ $$= 48 \ | \ 4$$ $$= 484$$

Example(3): Find the square of 23?

Solution: $$23 = 23 + 3 \ | \ 3^2$$ $$= 26 \times 2 \ | \ 9$$ $$= 52 \ | \ 9$$ $$= 529$$

Example(4): Find the square of 25?

Solution: $$25 = 25 + 5 \ | \ 5^2$$ $$= 30 \times 2 \ | \ 25$$ $$= 60 \ | \ 25$$ $$= 60 + 2 \ | \ 5$$ $$= 62 \ | \ 5$$ $$= 625$$

Square of Numbers between 30 to 39:

Example(1): Find the square of 30?

Solution: $$30 = 30 + 0 \ | \ 0^2$$ $$= 30 \times 3 \ | \ 0$$ $$= 90 \ | \ 0$$ $$= 900$$

Example(2): Find the square of 22?

Solution: $$32 = 32 + 2 \ | \ 2^2$$ $$= 34 \times 3 \ | \ 4$$ $$= 102 \ | \ 4$$ $$= 1024$$

Example(3): Find the square of 23?

Solution: $$33 = 33 + 3 \ | \ 3^2$$ $$= 36 \times 3 \ | \ 9$$ $$= 108 \ | \ 9$$ $$= 1089$$

Example(4): Find the square of 25?

Solution: $$36 = 36 + 6 \ | \ 6^2$$ $$= 42 \times 3 \ | \ 36$$ $$= 126 \ | \ 36$$ $$= 126 + 3 \ | \ 6$$ $$= 129 \ | \ 6$$ $$= 1296$$

Square of Numbers between 40 to 49:

Example(1): Find the square of 30?

Solution: $$40 = 40 + 0 \ | \ 0^2$$ $$= 40 \times 4 \ | \ 0$$ $$= 160 \ | \ 0$$ $$= 1600$$

Example(2): Find the square of 22?

Solution: $$45 = 45 + 5 \ | \ 5^2$$ $$= 50 \times 4 \ | \ 25$$ $$= 200 \ | \ 25$$ $$= 200 + 2 \ | \ 5$$ $$= 202 \ | \ 5$$ $$= 2025$$

Similarly, you can find the square of numbers till 99.

Square of Numbers between 100 to 109:

Example(1): Find the square of 100?

Solution: $$100 = 100 + 0 \ | \ 0^2$$ $$= 100 \times 10 \ | \ 0$$ $$= 1000 \ | \ 0$$ $$= 10000$$

Example(2): Find the square of 103?

Solution: $$103 = 103 + 3 \ | \ 3^2$$ $$= 106 \times 10 \ | \ 9$$ $$= 1060 \ | \ 9$$ $$= 10609$$

Example(3): Find the square of 109?

Solution: $$109 = 109 + 9 \ | \ 9^2$$ $$= 118 \times 10 \ | \ 81$$ $$= 1180 \ | \ 81$$ $$= 1180 + 8 \ | \ 1$$ $$= 1188 \ | \ 1$$ $$= 11881$$

Square of Numbers between 110 to 119:

Example(1): Find the square of 110?

Solution: $$110 = 110 + 0 \ | \ 0^2$$ $$= 110 \times 11 \ | \ 0$$ $$= 1210 \ | \ 0$$ $$= 12100$$

Example(2): Find the square of 112?

Solution: $$112 = 112 + 2 \ | \ 2^2$$ $$= 114 \times 11 \ | \ 4$$ $$= 1254 \ | \ 4$$ $$= 12544$$

Example(3): Find the square of 116?

Solution: $$116 = 116 + 6 \ | \ 6^2$$ $$= 122 \times 11 \ | \ 36$$ $$= 1342 \ | \ 36$$ $$= 1342 + 3 \ | \ 6$$ $$= 1345 \ | \ 6$$ $$= 13456$$

Square of Numbers between 120 to 129:

Example(1): Find the square of 120?

Solution: $$120 = 120 + 0 \ | \ 0^2$$ $$= 120 \times 12 \ | \ 0$$ $$= 1440 \ | \ 0$$ $$= 14400$$

Example(2): Find the square of 123?

Solution: $$123 = 123 + 3 \ | \ 3^2$$ $$= 126 \times 12 \ | \ 9$$ $$= 1512 \ | \ 9$$ $$= 15129$$

Example(3): Find the square of 127?

Solution: $$127 = 127 + 7 \ | \ 7^2$$ $$= 134 \times 12 \ | \ 49$$ $$= 1608 \ | \ 49$$ $$= 1608 + 4 \ | \ 9$$ $$= 1612 \ | \ 9$$ $$= 16129$$

Similarly, Here you can find the square of numbers till 199. But it is not the end at all, the square of any number can be found by using this method, Let's take some random examples to understand.

Example: Find the square of 255?

Solution: Here the number 255 is between 250 to 259, Hence $$255 = 255 + 5 \ | \ 5^2$$ $$= 260 \times 25 \ | \ 25$$ $$= 6500 \ | \ 25$$ $$= 6500 + 2 \ | \ 5$$ $$= 6502 \ | \ 5$$ $$= 65025$$

Example: Find the square of 325?

Solution: Here the number 325 is between 320 to 329, Hence $$325 = 325 + 5 \ | \ 5^2$$ $$= 330 \times 32 \ | \ 25$$ $$= 10560 \ | \ 25$$ $$= 10560 + 2 \ | \ 5$$ $$= 10562 \ | \ 5$$ $$= 105625$$

Important Points:

1. A perfect square always ends in 00, 1, 4, 6, 9, and 25.

2. The square of an Even Number is always an Even Number, and the square of an Odd Number is always an Odd Number.

3. If the last digit of any number is 0, then the square of that number always ends in 00.

4. If the last digit of any number is 1 or 9, then the square of that number always ends in 1.

5. If the last digit of any number is 2 or 8, then the square of that number always ends in 4.

6. If the last digit of any number is 3 or 7, then the square of that number always ends in 9.

7. If the last digit of any number is 4 or 6, then the square of that number always ends in 6.

8. If the last digit of any number is 5, then the square of that number always ends in 25.