Square Numbers Ending in 5
There is a quick mental math trick for squaring any number that ends in 5. The result always ends in 25, and the digits before 25 are obtained by multiplying the tens digit by the next integer.
This pattern works for all numbers ending in 5: 15, 25, 35, 45, and so on. It is a direct consequence of the algebraic identity (10a + 5)² = 100a(a + 1) + 25.
Learning this shortcut helps you compute squares mentally and quickly, which is useful in exams and competitive tests.
What is Square Numbers Ending in 5?
The rule: To square a number ending in 5:
(n5)² = n × (n + 1) followed by 25
Where:
- n = the digit(s) before 5
- Multiply n by (n + 1)
- Append 25 at the end
Examples:
- 15² → 1 × 2 = 2 → 225
- 25² → 2 × 3 = 6 → 625
- 35² → 3 × 4 = 12 → 1225
- 45² → 4 × 5 = 20 → 2025
- 75² → 7 × 8 = 56 → 5625
Methods
Algebraic proof:
- Let the number be 10a + 5 (a 2-digit number ending in 5).
- (10a + 5)² = 100a² + 100a + 25 = 100a(a + 1) + 25
- The term 100a(a + 1) gives the hundreds and higher places.
- The constant 25 gives the last two digits.
Steps for mental calculation:
- Take the digit(s) before 5. Call it n.
- Multiply: n × (n + 1).
- Write the product, followed by 25.
This works for 3-digit numbers too:
- 105² → 10 × 11 = 110 → 11025
- 115² → 11 × 12 = 132 → 13225
- 125² → 12 × 13 = 156 → 15625
Solved Examples
Example 1: Example 1: Square of 15
Problem: Find 15² mentally.
Solution:
- n = 1. n × (n+1) = 1 × 2 = 2.
- Append 25: 225
Answer: 15² = 225.
Example 2: Example 2: Square of 45
Problem: Find 45² mentally.
Solution:
- n = 4. n × (n+1) = 4 × 5 = 20.
- Append 25: 2025
Answer: 45² = 2025.
Example 3: Example 3: Square of 65
Problem: Find 65².
Solution:
- n = 6. n × (n+1) = 6 × 7 = 42.
- Append 25: 4225
Answer: 65² = 4225.
Example 4: Example 4: Square of 85
Problem: Find 85².
Solution:
- n = 8. n × (n+1) = 8 × 9 = 72.
- Append 25: 7225
Answer: 85² = 7225.
Example 5: Example 5: Square of 95
Problem: Find 95².
Solution:
- n = 9. n × (n+1) = 9 × 10 = 90.
- Append 25: 9025
Answer: 95² = 9025.
Example 6: Example 6: Three-digit number (105)
Problem: Find 105².
Solution:
- n = 10. n × (n+1) = 10 × 11 = 110.
- Append 25: 11025
Answer: 105² = 11025.
Example 7: Example 7: Square of 55
Problem: Find 55².
Solution:
- n = 5. n × (n+1) = 5 × 6 = 30.
- Append 25: 3025
Answer: 55² = 3025.
Example 8: Example 8: Square of 125
Problem: Find 125².
Solution:
- n = 12. n × (n+1) = 12 × 13 = 156.
- Append 25: 15625
Answer: 125² = 15625.
Example 9: Example 9: Algebraic verification
Problem: Prove 35² = 1225 using the identity.
Solution:
- 35 = 10(3) + 5
- (10 × 3 + 5)² = 100 × 3 × 4 + 25 = 1200 + 25 = 1225
- Direct: 35 × 35 = 1225 ✓
Answer: 35² = 1225. Verified.
Example 10: Example 10: Square of 5
Problem: Does the rule work for 5 itself?
Solution:
- n = 0. n × (n+1) = 0 × 1 = 0.
- Append 25: 025 = 25.
- 5² = 25 ✓
Answer: Yes, 5² = 25.
Real-World Applications
Where this is useful:
- Mental math competitions: Quickly squaring numbers ending in 5.
- Exam shortcuts: Saves time in board exams and competitive tests.
- Estimation: Getting approximate squares by rounding to the nearest multiple of 5.
- Understanding patterns: Recognising that perfect squares ending in 5 always end in 25.
Key Points to Remember
- To square a number ending in 5: multiply n by (n+1), then append 25.
- The algebraic basis: (10a + 5)² = 100a(a+1) + 25.
- This works for numbers of any size: 5, 15, 25, ..., 105, 115, ...
- The last two digits of the square are always 25.
- No perfect square ends in 2, 3, 7, or 8.
- All perfect squares ending in 5 must end in 25.
- This trick can be verified by regular multiplication.
Practice Problems
- Find 25² mentally.
- Find 75² using the shortcut.
- Find 115² using the trick.
- Prove that 145² = 21025.
- Find 995² mentally.
- Which is larger: 65² or 60 × 70?
Frequently Asked Questions
Q1. Why does this trick work?
Because (10a+5)² = 100a(a+1) + 25. The first part gives the digits before 25, and the 25 is always fixed.
Q2. Does it work for 3-digit numbers ending in 5?
Yes. For 135: n=13, 13×14=182, answer = 18225. Check: 135² = 18225.
Q3. Can all perfect squares end in 5?
Only squares of numbers ending in 5 end in 5 (specifically in 25). No other perfect square ends in 5.
Q4. What digits can a perfect square end in?
A perfect square can end in 0, 1, 4, 5, 6, or 9 only. It can NEVER end in 2, 3, 7, or 8.
Q5. Is 65² the same as 60 × 70 + 25?
Yes! n(n+1) for n=6 gives 42, and 42 × 100 + 25 = 4225. Also, 60 × 70 = 4200, and 4200 + 25 = 4225.
Q6. Can I use this for numbers not ending in 5?
No, this specific trick only works for numbers ending in 5. For other numbers, use standard multiplication or other identities.
