**DEALING WITH SQUARES OF FIVE
**

__Table of Multiples of Five Squares__ (M5.Sq)

^{2} = 0’25

**15 ^{2} = 2’25**

**25 ^{2} = 6’25**

**35 ^{2} = 12’25**

**45 ^{2} = 20’25**

**55 ^{2} = 30’25**

**65 ^{2} = 42’25**

**75 ^{2} = 56’25**

**85 ^{2} = 72’25**

**95 ^{2} = 90’25**

**common things**” about these following groups;

__Group 1__

^{2} = __2__’25

**35 ^{2} = 12’25**

**65 ^{2} = 42’25**

**85 ^{2} = 72’25**

__Group 2__

**
**

**05 ^{2} = 0’25 **(sometimes, not included)

**45 ^{2} = 20’25**

**55 ^{2} = 30’25**

**95 ^{2} = 90’25**

__Group 3__

^{2} = __6__’25

**75 ^{2} = 56’25**

__Take Note____:__

1) The squares of

**15, 35, 65**and

**85**, all end up with

**…**

__2__25

**5, 45, 55** and **95**, all end up with **… 025**

**… 6’25**

**all squares** of numbers having a **last digit of 5**, will all, end up with **…25**.

But the underlined numbers will give us a ‘hint’ of what the missing digits might be.

**√ 403,225**

__Step 1:__

Use the basic instructions of Square Edging (Review SE Telegram: http://easysqrtsforkids.blogspot.com/2010/08/se-telegram_13.html )

__2__’25

... **6** ..**_** .. **5**

.. **N : 1,3,6,8**

__Step 2__:

Create a Parameter Checker (**P-Chk**)

**65: 42’25↓**

**4↓ N : {0, 1, 2, 3, 4}**

__2__’25).

__Step 3__

Underline the 1 and 3, indicating that 6 and 8 are eliminated:

**65: 42’25↓**

**4↓ N : {0, 1, 2, 3, 4}**

__Step 4__

Create A Square Root Locator

**↓ 615** ..... **635 ↑**

**42’25 \**

**39’06 / ↑**

__36 ..... __

__78’25 /__ 2

**39’12**

**√ 403,225 = 635**

**MSM-2 **seems to be simple when dealing with “squares of 5”in five or six digits. All you have to do is to determine which group the missing “middle-digit” belongs and slim down the possibility by using the P-Chk.

__
__

__“SQUARES OF FIVE” IN EIGHT DIGITS __

__Case No. 1__

**√ 21,949,225**

__Step 1__:

Use the SE telegram procedure as initial instructions:

**√ 21’94’9 2’25**

.... **4** ........... **5**

**N : 1, 3, 6, 8**

__Step 2:__

Create A P-Chk

**P : 21’94 **

**45: 20’25 ↑**

**5↑ M : 5, 6, 7, 8, 9**

**4 515, 4535, 4565, 4585, 4615, 4635, 4665, 4685, 4715, 4735, 4765, 4785, 4815, 4835, 4865, 4885, 4915, 4935, 4965 **and

**4**and only one of the 20 possible square root is the true square root

__9__85

__Step 3:__

To slim down the possibilities, we must use the ^{st} Square

**25 **.....

**22’56 \ ↓**

__20’25 /__

__45’25 / __2

**22’62**

__Modify Step 2__:

1) As a rule, underline 5, 6, 7(or draw a box, enclosing 5, 6 and 7 of the P-Chk )

2) Insert the notation “/_750/” on the second line of P-Chk . The purpose of this is determine the limit of set of possible square roots

**45: 20’25 ↑ /4750/**

**5↑ M : 5, 6, 7, 8, 9**

__Step 4: __

On the next column, write down the set of possible square roots

...

**↓**........

**↑**

**4615** ..... **4735**

**4585 **..... **4715**

**4565 **..... **4685**

**4535 **..... **4665**

**4515 **..... **4635**

**4515**, write it down by starting from the bottom part, going upward up to **4615 **(five elements at each set). Put a ‘down arrow’ symbol on top of the first set

**4635 **up to **4735**, following the same procedures. This time, put an ‘up arrow’.

**4750**/ on the second line of the P-Chk give us the idea that the ‘**upper limit**’ only include **4735**. Therefore, 4765 and above, are eliminated.

__Step 5: __

Use the ^{nd} Square

**22’56 \**

**21’40 / ↑**

__20’25__

__42’81 /__ 2

**21’40**

**4715**

**4685**

**4665**

**4635**

__Important:__

Enclose and put an ‘**up arrow’** ↑ for {**4685, 4715, and 4735**} while a ‘**down arrow’**↓ for {**4635, 4665 and 4685}**

...

**↑**

**4735**

**4715 **

**4685 **

...

**↓**

**4685 **

**4665**

**4635**

__Step 5__:

Continue the process of averaging the square values ( ^{rd} Square

**22’56**

**21’98 \ ↓**

__21’40 /__

__43’96 /__ 2

**21’98**

**down arrow ↓ eliminates - 4685, 4715, and 4735**. There are still 3 remaining square roots:

**↑ / 4685 **

....** \ 4665 \ **

......** 4635 / ↓**

__Step 6:__

Continue the averaging and elimination process (4^{th} square root locator)

**21’98 \**

**21’69 / ↑**

__21’40__

__43’38 /__ 2

**21’69**

**4635 is eliminated**.

**↓4665** ....... **4685↑**

**
**

**21’98 \**

**21’83 / ↑**

__21’69__

__43’67 /__ 2

**21’83**

**up arrow** indicates that the **true square root is 4685**

**√ 21,949,225 = 4,685**

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