Important Notes

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Remember:-

1. Last Digit of any [Even Number] ^ 4n = 6

Last Digit of any [Odd Number] ^ 4n = 1

There is exception to above rule. This rule is not applicable for number ending with 0, 1, 5 and 6 because cyclicity of these numbers is 1.

2. [25]^n always ends with 25.

[76]^n always end with 76.

3. Perfect Number: -

A number is a perfect number when sum of all its factors excluding number itself equals to number.

Eg. 6 Factors à 1, 2, 3 and 6

Ie 1 + 2 + 3 = 6

28, 496 and 8128 are few of such numbers.

4. Consider a number formed with digit a and b ie ab then number formed by reversing the digit with be ba then

ab = 10 a+ b

ba = 10 b + a

=> ab + ba = 11 (a + b) & ab – ba = 9 (a-b)

5. Product of three consecutive numbers is always divisible by 3. ie number of the form

(p-1)p(p+1) is always divisible by 3.