Finding Squares 51²-59², 151²-159², 251²-259²

Here's a smart mental maths trick to square numbers that fall just after a multiple of 50 — like 51², 152², 357², 658², and so on. The beauty of this pattern is that it extends to any series ending with digits 1-9 where the leading part is a multiple of 5. In other words, this trick works beautifully for numbers in the range 51² to 59², 151² to 159², 251² to 259², 351² to 359², and so on.

Before you try this trick...

Make sure you're comfortable with finding the square of numbers ending in 5 — because that's where the a² part comes from.

Try our blog Finding the square of numbers ending in 5 and understand it fully before moving on.

The steps are...

Left to Right

LTR

This method follows a left-to-right calculation flow.

Finding the value of 52²

5252

Split 52 as a = 5 and b = 2

Here, a always ends with 5.

The steps are:

The answer has two parts:

Part I Finding a² (a 5) b

2 = 27

Part II Finding b²

= 04

Join the above two parts, to get the answer:

2704

52² = 2704

Solution for all the given numbers

Steps	Finding 52²
1	Split `52` as `a = 5` and `b = 2`
2	Part I = a² (a 5) b = 5² (5 5) 2 = 25 2 = 27
3	Part II = b² = 2² = 04 (Must be in 2-digit format)
4	Join the two parts 27 and 04 to get the solution: 2704

Steps	Finding 54²
1	Split `54` as `a = 5` and `b = 4`
2	Part I = a² (a 5) b = 5² (5 5) 4 = 25 4 = 29
3	Part II = b² = 4² = 16
4	Join the two parts 29 and 16 to get the solution: 2916

Steps	Finding 56²
1	Split `56` as `a = 5` and `b = 6`
2	Part I = a² (a 5) b = 5² (5 5) 6 = 25 6 = 31
3	Part II = b² = 6² = 36
4	Join the two parts 31 and 36 to get the solution: 3136

Steps	Finding 58²
1	Split `58` as `a = 5` and `b = 8`
2	Part I = a² (a 5) b = 5² (5 5) 8 = 25 8 = 33
3	Part II = b² = 8² = 64
4	Join the two parts 33 and 64 to get the solution: 3364

Steps	Finding 151²
1	Split `151` as `a = 15` and `b = 1`
2	Part I = a² (a 5) b = 15² (15 5) 1 = 225 3 = 228
3	Part II = b² = 1² = 01 (Must be in 2-digit format)
4	Join the two parts 228 and 01 to get the solution: 22801

Steps	Finding 153²
1	Split `153` as `a = 15` and `b = 3`
2	Part I = a² (a 5) b = 15² (15 5) 3 = 225 9 = 234
3	Part II = b² = 3² = 09 (Must be in 2-digit format)
4	Join the two parts 234 and 09 to get the solution: 23409

Steps	Finding 155²
1	Split `155` as `a = 15` and `b = 5`
2	Part I = a² (a 5) b = 15² (15 5) 5 = 225 15 = 240
3	Part II = b² = 5² = 25
4	Join the two parts 240 and 25 to get the solution: 24025

Steps	Finding 157²
1	Split `157` as `a = 15` and `b = 7`
2	Part I = a² (a 5) b = 15² (15 5) 7 = 225 21 = 246
3	Part II = b² = 7² = 49
4	Join the two parts 246 and 49 to get the solution: 24649

Steps	Finding 252²
1	Split `252` as `a = 25` and `b = 2`
2	Part I = a² (a 5) b = 25² (25 5) 2 = 625 10 = 635
3	Part II = b² = 2² = 04 (Must be in 2-digit format)
4	Join the two parts 635 and 04 to get the solution: 63504

Steps	Finding 254²
1	Split `254` as `a = 25` and `b = 4`
2	Part I = a² (a 5) b = 25² (25 5) 4 = 625 20 = 645
3	Part II = b² = 4² = 16
4	Join the two parts 645 and 16 to get the solution: 64516

Steps	Finding 256²
1	Split `256` as `a = 25` and `b = 6`
2	Part I = a² (a 5) b = 25² (25 5) 6 = 625 30 = 655
3	Part II = b² = 6² = 36
4	Join the two parts 655 and 36 to get the solution: 65536

Steps	Finding 258²
1	Split `258` as `a = 25` and `b = 8`
2	Part I = a² (a 5) b = 25² (25 5) 8 = 625 40 = 665
3	Part II = b² = 8² = 64
4	Join the two parts 665 and 64 to get the solution: 66564

Steps	Finding 354²
1	Split `354` as `a = 35` and `b = 4`
2	Part I = a² (a 5) b = 35² (35 5) 4 = 1225 28 = 1253
3	Part II = b² = 4² = 16
4	Join the two parts 1253 and 16 to get the solution: 125316

Steps	Finding 356²
1	Split `356` as `a = 35` and `b = 6`
2	Part I = a² (a 5) b = 35² (35 5) 6 = 1225 42 = 1267
3	Part II = b² = 6² = 36
4	Join the two parts 1267 and 36 to get the solution: 126736

Steps	Finding 453²
1	Split `453` as `a = 45` and `b = 3`
2	Part I = a² (a 5) b = 45² (45 5) 3 = 2025 27 = 2052
3	Part II = b² = 3² = 09 (Must be in 2-digit format)
4	Join the two parts 2052 and 09 to get the solution: 205209

Steps	Finding 457²
1	Split `457` as `a = 45` and `b = 7`
2	Part I = a² (a 5) b = 45² (45 5) 7 = 2025 63 = 2088
3	Part II = b² = 7² = 49
4	Join the two parts 2088 and 49 to get the solution: 208849

Steps	Finding 552²
1	Split `552` as `a = 55` and `b = 2`
2	Part I = a² (a 5) b = 55² (55 5) 2 = 3025 22 = 3047
3	Part II = b² = 2² = 04 (Must be in 2-digit format)
4	Join the two parts 3047 and 04 to get the solution: 304704

Steps	Finding 556²
1	Split `556` as `a = 55` and `b = 6`
2	Part I = a² (a 5) b = 55² (55 5) 6 = 3025 66 = 3091
3	Part II = b² = 6² = 36
4	Join the two parts 3091 and 36 to get the solution: 309136

Steps	Finding 654²
1	Split `654` as `a = 65` and `b = 4`
2	Part I = a² (a 5) b = 65² (65 5) 4 = 4225 52 = 4277
3	Part II = b² = 4² = 16
4	Join the two parts 4277 and 16 to get the solution: 427716

Steps	Finding 655²
1	Split `655` as `a = 65` and `b = 5`
2	Part I = a² (a 5) b = 65² (65 5) 5 = 4225 65 = 4290
3	Part II = b² = 5² = 25
4	Join the two parts 4290 and 25 to get the solution: 429025

Steps	Finding 756²
1	Split `756` as `a = 75` and `b = 6`
2	Part I = a² (a 5) b = 75² (75 5) 6 = 5625 90 = 5715
3	Part II = b² = 6² = 36
4	Join the two parts 5715 and 36 to get the solution: 571536

Steps	Finding 857²
1	Split `857` as `a = 85` and `b = 7`
2	Part I = a² (a 5) b = 85² (85 5) 7 = 7225 119 = 7344
3	Part II = b² = 7² = 49
4	Join the two parts 7344 and 49 to get the solution: 734449

Steps	Finding 959²
1	Split `959` as `a = 95` and `b = 9`
2	Part I = a² (a 5) b = 95² (95 5) 9 = 9025 171 = 9196
3	Part II = b² = 9² = 81
4	Join the two parts 9196 and 81 to get the solution: 919681

Steps	Finding 957²
1	Split `957` as `a = 95` and `b = 7`
2	Part I = a² (a 5) b = 95² (95 5) 7 = 9025 133 = 9158
3	Part II = b² = 7² = 49
4	Join the two parts 9158 and 49 to get the solution: 915849

Introduction

Before you try this trick...

Choose a number

The steps are...

Solution for all the given numbers