Squares of numbers ending in 4 & 6 by Bhārati Kṛṣṇa Tīrtha
A number, when multiplied by the number itself the product obtained, is called "Square of that number ". We will come to know the squares for numbers ending in a particular digit 4 , 6.
Squares of Numbers ending in 4[1]
The specific rule to be followed to find the square of a number ending in 4 is explained through the below examples.
Example : 342
Step1 : We will find the square of its nearest number ending in 5. 35 (number ending in 5) is closer to 34.
Here the number 35 ending in 5, we will use एकाधिकेन पूर्वेण sūtra to find its square.
| Left Hand Side (LHS) | Right Hand Side (RHS) |
|---|---|
| 3 | 5 |
| Previous digit of 5 in 35 is 3
one more than 3 is 4 |
Square of 5 = 25 |
| 3 X 4 = 12 | |
| 12 | 25 |
352 = 1225
Step 2 : Add the number with the nearest number ending in 5. 34 + 35 = 69
Step 3 : Subtract the square of the nearest number ending in 5 ( value obtained in Step 1) with the value obtained in Step 2. 1225 - 69 = 1156
Answer : 342 = 1156
Example : 6242
Step1 : We will find the square of its nearest number ending in 5. 625 (number ending in 5) is closer to 624.
Here the number 625 ending in 5, we will use एकाधिकेन पूर्वेण sūtra to find its square.
| Left Hand Side (LHS) | Right Hand Side (RHS) |
|---|---|
| 62 | 5 |
| Previous digit of 5 in 625 is 62
one more than 62 is 63 |
Square of 5 = 25 |
| 62 X 63 = 3906* | |
| 3906 | 25 |
6252 = 390625
*To find the product of 62 X 63 we use ऊर्ध्वतिर्यग्भ्याम् (Ūrdhvatiryagbhyām)
| Left Column | Right Column | |
|---|---|---|
| First Digit | 6 | 2 |
| Second Digit | 6 | 3 |
Step 1: Vertically multiply the two digits in the right column - 2 X 3 =6
Step 2: Cross multiply the first digit of the right column with the second digit of the left column. Second digit of the right column with the first digit of the left column and add the two products. - (2 X 6) + (3 X 6) = 12 + 18 = 30
Step 3: Vertically multiply the two digits in the left column - 6 X 6 = 36
Step 4 : Put the above values got from each step in the below table.
| Step 3 | Step 2 | Step 1 |
| 36 | 30 | 6 |
| 36 + 3 | 0 | 6 |
| 39 | 0 | 6 |
62 X 63 = 3906
Step 2 : Add the number with the nearest number ending in 5. 624 + 625 = 1249
Step 3 : Subtract the square of the nearest number ending in 5 ( value obtained in Step 1) with the value obtained in Step 2. 390625 - 1249 = 389376
Answer : 6242 = 389376
Squares of Numbers ending in 6
The specific rule to be followed to find the square of a number ending in 6 is explained through the below examples.
Example : 362
Step1 : We will find the square of its nearest number ending in 5. 35 (number ending in 5) is closer to 36.
Here the number 35 ending in 5, we will use एकाधिकेन पूर्वेण sūtra to find its square.
| Left Hand Side (LHS) | Right Hand Side (RHS) |
|---|---|
| 3 | 5 |
| Previous digit of 5 in 35 is 3
one more than 3 is 4 |
Square of 5 = 25 |
| 3 X 4 = 12 | |
| 12 | 25 |
352 = 1225
Step 2 : Add the number with the nearest number ending in 5. 36 + 35 = 71
Step 3 : Add the square of the nearest number ending in 5 ( value obtained in Step 1) with the value obtained in Step 2. 1225 + 71 = 1296
Answer : 362 = 1156
Example : 2462
Step1 : We will find the square of its nearest number ending in 5. 245 (number ending in 5) is closer to 246.
Here the number 245 ending in 5, we will use एकाधिकेन पूर्वेण sūtra to find its square.
| Left Hand Side (LHS) | Right Hand Side (RHS) |
|---|---|
| 24 | 5 |
| Previous digit of 5 in 245 is 24
one more than 24 is 25 |
Square of 5 = 25 |
| 24 X 25 = 600* | |
| 600 | 25 |
2452 = 60025
*To find the product of 24 X 25 we use ऊर्ध्वतिर्यग्भ्याम् (Ūrdhvatiryagbhyām)
| Left Column | Right Column | |
|---|---|---|
| First Digit | 2 | 4 |
| Second Digit | 2 | 5 |
Step 1: Vertically multiply the two digits in the right column - 4 X 5 = 20
Step 2: Cross multiply the first digit of the right column with the second digit of the left column. Second digit of the right column with the first digit of the left column and add the two products. - (4 X 2) + (5 X 2) = 8 + 10 = 18
Step 3: Vertically multiply the two digits in the left column - 2 X 2 = 4
Step 4 : Put the above values got from each step in the below table.
| Step 3 | Step 2 | Step 1 |
| 4 | 18 | 20 |
| 4 | 18+2 | 0 |
| 4 | 20 | 0 |
| 4+2 | 0 | 0 |
| 6 | 0 | 0 |
24 X 25 = 600
Step 2 : Add the number with the nearest number ending in 5. 245 + 246 = 491
Step 3 : Add the square of the nearest number ending in 5 ( value obtained in Step 1) with the value obtained in Step 2. 60025 + 491 = 60516
Answer : 2462 = 60516
See Also
4 और 6 से समाप्त होने वाली संख्याओं का वर्ग - भारती कृष्ण तीर्थ
References
- ↑ Singhal, Vandana (2007). Vedic Mathematics For All Ages - A Beginners' Guide. Delhi: Motilal Banarsidass. pp. 204–209. ISBN 978-81-208-3230-5.
