Finding the square of numbers ending in 5

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.

The steps are...

Finding the value of 35²...
Step 1:
Ignoring the unit digit (or the ones digit) 5, take the remaining digits as a number. Let this number be n.	The chosen integer is 35 and n = 3
35² is ????
Step 2:
Multiply the number n by its successor n + 1. This product forms the beginning digits of the required square value.	Here n = 3 and n + 1 = 4
The product is n ✕ (n + 1)	The product is 3 ✕ 4 = 12
35² is 12??
Step 3:
After the above product, append the number 25 to get the required square value.	After the above product 12, append the number 25 to get the required square value, that is 1225.
35² is 1225

Quick application of the steps on all the given numbers

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:
15²	1	1 ✕ 2 = 2	225
25²	2	2 ✕ 3 = 6	625
35²	3	3 ✕ 4 = 12	1225
45²	4	4 ✕ 5 = 20	2025
55²	5	5 ✕ 6 = 30	3025
65²	6	6 ✕ 7 = 42	4225
75²	7	7 ✕ 8 = 56	5625
85²	8	8 ✕ 9 = 72	7225
95²	9	9 ✕ 10 = 90	9025
105²	10	10 ✕ 11 = 110	11025

In equation form

The logic of this trick is nothing but

n² = (n ✕ (n + 1)) ✕ 10 + 25

where "n" is a number ending in 5.

Some bigger numbers ending in 5

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.

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

Use the Times Tables Reference to learn and memorize.

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:
115²	11	11 ✕ 12 = 11² + 11 = 132 or 11 ✕ 12 = 12² - 12 = 132	13225
125²	12	12 ✕ 13 = 12² + 12 = 156 or 12 ✕ 13 = 13² - 13 = 156	15625
135²	13	13 ✕ 14 = 13² + 13 = 182 or 13 ✕ 14 = 14² - 14 = 182	18225
145²	14	14 ✕ 15 = 14² + 14 = 210 or 14 ✕ 15 = 15² - 15 = 210	21025
155²	15	15 ✕ 16 = 15² + 15 = 240 or 15 ✕ 16 = 16² - 16 = 240	24025
165²	16	16 ✕ 17 = 16² + 16 = 272 or 16 ✕ 17 = 17² - 17 = 272	27225
175²	17	17 ✕ 18 = 17² + 17 = 306 or 17 ✕ 18 = 18² - 18 = 306	30625
185²	18	18 ✕ 19 = 18² + 18 = 342 or 18 ✕ 19 = 19² - 19 = 342	34225
195²	19	19 ✕ 20 = 19² + 19 = 380 or 19 ✕ 20 = 20² - 20 = 380	38025
205²	20	20 ✕ 21 = 20² + 20 = 420 or 20 ✕ 21 = 21² - 21 = 420	42025
215²	21	21 ✕ 22 = 21² + 21 = 462 or 21 ✕ 22 = 22² - 22 = 462	46225
225²	22	22 ✕ 23 = 22² + 22 = 506 or 22 ✕ 23 = 23² - 23 = 506	50625
235²	23	23 ✕ 24 = 23² + 23 = 552 or 23 ✕ 24 = 24² - 24 = 552	55225
245²	24	24 ✕ 25 = 24² + 24 = 600 or 24 ✕ 25 = 25² - 25 = 600	60025
255²	25	25 ✕ 26 = 25² + 25 = 650 or 25 ✕ 26 = 26² - 26 = 650	65025
265²	26	26 ✕ 27 = 26² + 26 = 702 or 26 ✕ 27 = 27² - 27 = 702	70225
275²	27	27 ✕ 28 = 27² + 27 = 756 or 27 ✕ 28 = 28² - 28 = 756	75625
285²	28	28 ✕ 29 = 28² + 28 = 812 or 28 ✕ 29 = 29² - 29 = 812	81225
295²	29	29 ✕ 30 = 29² + 29 = 870 or 29 ✕ 30 = 30² - 30 = 870	87025
305²	30	30 ✕ 31 = 30² + 30 = 930 or 30 ✕ 31 = 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.