Can you find 101², 105², or 110² without long multiplication? Yes. Numbers from 101 to 110 follow a useful square-number pattern that makes mental math faster and easier to check.
In this Maths Tricks guide, you will learn how to square numbers just above 100. The method uses the identity (100 + n)2, where n is the small amount added to 100.
The Quick Pattern: 101² to 110²
For a number of the form 100 + n, square 100 first, add 200 times n, and then add n squared:
The pattern is easy to see in the answers. The leading part increases by 2 each time, while the final two digits are the squares of 1 through 10:
The Main Shortcut: Use 100 as the Base
Write the number as 100 + n. Then use the square expansion:
This comes from the algebra rule (a + b)2 = a2 + 2ab + b2. Because the difference from 100 is small, the calculation can often be done mentally.
Example 1: Find 101²
101 is 100 + 1, so use n = 1:
Example 2: Find 105²
105 is 100 + 5, so use n = 5:
Example 3: Find 109²
109 is 100 + 9, so use n = 9:
The Complete 101–110 Square Table
| Number | Write As | Square |
|---|---|---|
| 101 | 100 + 1 | 10,201 |
| 102 | 100 + 2 | 10,404 |
| 103 | 100 + 3 | 10,609 |
| 104 | 100 + 4 | 10,816 |
| 105 | 100 + 5 | 11,025 |
| 106 | 100 + 6 | 11,236 |
| 107 | 100 + 7 | 11,449 |
| 108 | 100 + 8 | 11,664 |
| 109 | 100 + 9 | 11,881 |
| 110 | 100 + 10 | 12,100 |
Why the Pattern Works
Every number from 101 to 110 is close to 100. When you expand (100 + n)2, the middle term is 200n, so each step adds about 200 to the square. The final adjustment is n2, which creates the ending pattern 01, 04, 09, 16, 25, 36, 49, 64, 81, and 100.
Common Mistakes to Avoid
Do not write only the last two digits and forget the leading part. For example, 107² does not equal 49; the full answer is 11,449. Also remember that 110 is 100 + 10, so the final adjustment is 10² = 100.
Use a quick estimate to check your answer. Since 105 is a little larger than 100, its square should be a little larger than 10,000. The answer 11,025 is reasonable, while an answer such as 1,025 is not.
Practice Problems
Try these without a calculator:
- 102²
- 104²
- 106²
- 108²
- 110²
Answer Key
- 102² = 10,404
- 104² = 10,816
- 106² = 11,236
- 108² = 11,664
- 110² = 12,100
Final Takeaway
To square a number from 101 to 110, write it as 100 + n and use:
This shortcut makes the 101–110 square table easier to remember and helps you solve mental-math questions more confidently. Save this Maths Tricks guide for homework, quiz review, and fast calculations.
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