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A two digit number

You are aware of two digit numbers. A two digit number is written in the form ab where ab = 10×a + b. Similarly as we would write 21 = 2×10 + 1. In this post I will teach you how to play with two digit numbers. I will teach fast tricks to do multiplication and finding squares.

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Multiplication

You are aware of the formula (a + b)(c + d) = a×c + a×d + b×c + b×d. If you have two digit numbers kl and mn. Then you can represent kl as 10×k + l and mn as 10×m + n.

The product of kl and mn can be represented as

(10k + l)(10m + n) = 100×k×m + 10(k×n + l×m) + l×n

Example

21 × 65

= 100×2×6 + 10(2×5 + 1×6) + 1×5

= 1200 + 10(16) + 5

= 1200+160+5

= 1365
**Steps to multiply two numbers**
- Multiply the digits at unit place
- Multiply the two sets of digits at unit place and digit at tens place. Add. Multiply the result by 10.

- Multiply the two digits at tens place. Multiply the result by 100.

- Add the result obtained in steps 1,2 and 3.

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Square

You know that the square of a number is product of a number with itself. So, if the number is kl then the square can be represented as

(10k + l)(10k + l) = 100×k×k + 10(k×l + l×k) + l×l

= 100×k

^{2} + 2×10×(k×l) + l

^{2}
You can obtain the above formula easily by using the expansion

(a + b)

^{2} = a

^{2} + 2ab + b

^{2}
**Example**

15×15

= 100×1 + 2×10×5 + 25

= 100 + 100 + 25

= 225
**Steps to square a two digit number**
- Square the digit at unit place
- Multiply the two digits at unit place and at tens place. Multiply the result by 20.

- Square the digit at tens place and multiply by 100.

- Add the result obtained in steps 1,2 and 3.

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Squaring a two digit number ending with 5

When the number ends with 5 it can be written as k5.

Its square is 100×k

^{2} + 2×10×(k×5) + 5

^{2}
Which can be written as 100×k

^{2} + 100×k + 25

= 100×(k

^{2}+k) + 25

= 100×(k(k+1)) + 25

*Rule: Find the product of digit at tens place and (digit at tens place + 1). Place 25 after it.*