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Finding Squares of Numbers Ending in 5

High School Student reading a maths blog

Introduction

This is a simple trick to quickly find the squares of numbers that end in 5. In other words, those numbers whose unit digit (or ones digit) is the number 5. This is just a specific case of the trick Quickly multiply when the unit digits add up to 10

Choose a number

The Trick is...

Finding the value of 352

35✕35
The steps are:
The answer has two parts:
Part 1
Multiply 3 by (3 plus 1)
3 ✕ 4 = 12
Part 2
Enter 25.  
25 = 52
Join the above two parts, to get the answer:
1225
352 = 1225

The Trick in detail

  • Step 1: Ignoring the unit digit 5, take the remaining digits as a number.
  • Step 2: Multiply the number by its successor.
  • Step 3: Append 25 to the product from Step 2 to get the required square value.
 Step 1:Step 2:Step 3:
1521
1
✕
2
=
2
225
2522
2
✕
3
=
6
625
3523
3
✕
4
=
12
1225
4524
4
✕
5
=
20
2025
5525
5
✕
6
=
30
3025
6526
6
✕
7
=
42
4225
7527
7
✕
8
=
56
5625
8528
8
✕
9
=
72
7225
9529
9
✕
10
=
90
9025
105210
10
✕
11
=
110
11025
115211
11
✕
12
=
132
13225
125212
12
✕
13
=
156
15625
135213
13
✕
14
=
182
18225
145214
14
✕
15
=
210
21025
155215
15
✕
16
=
240
24025
165216
16
✕
17
=
272
27225

In equation form

The logic of this trick is nothing but

n2 = (n ✕ (n + 1)) ✕ 100 + 25

where n is a number ending in 5.

What you need to know before you start...

You need to master Multiplication Tables (1 to 20) and Squares (1 to 30). Memorizing these essential concepts is crucial for any mathematics student. Not only will it benefit this blog post, but it will also prove invaluable for numerous other topics covered on our blogs.

Our specially crafted worksheets on Multiplication and Squares will help you solidify this knowledge.

Use the Times Tables Reference and Squares and Cubes Reference to learn and memorize.

This Trick on Bigger numbers

Let's use the same trick to quickly find the squares of some bigger numbers from the series 115, 125, 135, 145... up to 305.

  • Step 1: Ignoring the unit digit 5, take the remaining digits as a number.
  • Step 2: Multiply the number by its successor using the trick.†
  • Step 3: Append 25 to the product from Step 2 to get the required square value.
 Step 1:Step 2:†Step 3:
115211
11 ✕ 12

112 + 11 = 132
Or
122 - 12 = 132
13225
125212
12 ✕ 13

122 + 12 = 156
Or
132 - 13 = 156
15625
135213
13 ✕ 14

132 + 13 = 182
Or
142 - 14 = 182
18225
145214
14 ✕ 15

142 + 14 = 210
Or
152 - 15 = 210
21025
155215
15 ✕ 16

152 + 15 = 240
Or
162 - 16 = 240
24025
165216
16 ✕ 17

162 + 16 = 272
Or
172 - 17 = 272
27225
175217
17 ✕ 18

172 + 17 = 306
Or
182 - 18 = 306
30625
185218
18 ✕ 19

182 + 18 = 342
Or
192 - 19 = 342
34225
195219
19 ✕ 20

192 + 19 = 380
Or
202 - 20 = 380
38025
205220
20 ✕ 21

202 + 20 = 420
Or
212 - 21 = 420
42025
215221
21 ✕ 22

212 + 21 = 462
Or
222 - 22 = 462
46225
225222
22 ✕ 23

222 + 22 = 506
Or
232 - 23 = 506
50625
235223
23 ✕ 24

232 + 23 = 552
Or
242 - 24 = 552
55225
245224
24 ✕ 25

242 + 24 = 600
Or
252 - 25 = 600
60025
255225
25 ✕ 26

252 + 25 = 650
Or
262 - 26 = 650
65025
265226
26 ✕ 27

262 + 26 = 702
Or
272 - 27 = 702
70225
275227
27 ✕ 28

272 + 27 = 756
Or
282 - 28 = 756
75625
285228
28 ✕ 29

282 + 28 = 812
Or
292 - 29 = 812
81225
295229
29 ✕ 30

292 + 29 = 870
Or
302 - 30 = 870
87025
305230
30 ✕ 31

302 + 30 = 930
Or
312 - 31 = 930
93025

Our worksheets

Once you've learnt this trick, please do visit Squares and Cubes worksheet pages and put your learnings into practice.