1.2.8 Squares Ending in 5 Trick

Any number ending in 5, when squared, follows a simple pattern. Here’s the derivation:

Let a5 represent any number ending in 5 (a could be any integer):

(a5)² = (10a + 5)²

= 100a² + 100a + 25

= 100a(a + 1) + 25

The Rule

For any number ending in 5:

1. Last two digits are always 25
2. Remaining digits = leading digit(s) × (leading digit(s) + 1)

Example: 85²

Step	Calculation	Result
Tens/Ones	Always	25
Thousands/Hundreds	8 × (8 + 1) = 8 × 9	72

Answer: 7225

Example: 115²

Step	Calculation	Result
Tens/Ones	Always	25
Rest of Answer	11 × (11 + 1) = 11 × 12	132

Answer: 13225

Advanced Example: 15⁴

This example shows how to compute 15⁴ by applying the squares ending in 5 trick twice:

Step 1: Find 15²

Step	Calculation	Result
Tens/Ones	Always	25
Thousands/Hundreds	1 × (1 + 1) = 1 × 2	2

15² = 225

Step 2: Find 225²

Step	Calculation	Result
Tens/Ones	Always	25
Rest of Answer	22 × 23 = 11 × 46	506

225² = 50625

Tip: In the above trick we also used the double/half trick (22 × 23 = 11 × 46) and could use the 11’s trick. This shows that for some problems, using multiple tricks might be necessary!

More Examples

Number	Calculation	Answer
25²	2 × 3 = 6, append 25	625
35²	3 × 4 = 12, append 25	1225
45²	4 × 5 = 20, append 25	2025
55²	5 × 6 = 30, append 25	3025
65²	6 × 7 = 42, append 25	4225
75²	7 × 8 = 56, append 25	5625
95²	9 × 10 = 90, append 25	9025
105²	10 × 11 = 110, append 25	11025

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Problem Set 1.2.8

Practice these problems. Type your answer and press Enter to check:

Basic Squares Ending in 5

Larger Squares

Applied Problems

Challenge Problems