**Shortcut 77:**

Divisibility test

*Divisibleby Rule*

*2 Last digit of a number should be 0,2,4,6, or 8*

*3 Sum of the digit should be divisible by 3*

*4 Last two digits of a number should be divisible by 4*

*5 Last digit of a number should be 0 or 5*

*6 Number should be divisible by 2 and 3*

*8 Last three digit of a number shoul be divisible by 8*

*9 Sum of the digit should be divisible by 9*

*10 Last three digit of a number shoul be 0*

*11 Difference between sum of digits in even places shoud be 0 or 11*

Question:

Which of the following numbers is divisible by both 5 and 9?

1115, 11115, 111115, 1111115

Answer:

11115 – Its is divisible by 5 and 9

**Shortcut 78:**

Finding unit digit

*Base Value Power Value*

*Unit Place Unit Place*

* 0 0*

* 1 1*

* 5 5*

* 6 6*

Question:

Find the units place in the result of the expression:

240234 + 345344 x 1011009 – 36211

Answer:

Unit’s place of 240234 = 0

Unit’s place of 345344 = 5

Unit’s place of 1011009 = 1

Unit’s place of 36211 = 6

Units place of the expression

= 0 + 5 x 1 – 6 = 9

(When subtracting 5 from 6 in unit’s place we will borrow 1 for 5, 15 -6 = 9)

**Shortcut 79:**

Finding unit digit

* Odd Even*

* 4 4 6*

* 9 9 1*

Question:

Find the unit’s digit of 3449 and 4934.

Answer:

3449 = 4 power odd number – 4 in unit’s place

So, unit digit of 3449 = 4

4934 = 9 power even number – 1 in unit’s place

So, unit’s digit of 4934 = 1

**Shortcut 80:**

Finding unit digit

Question:

Find the unit digit of the expression

1211 + 1312 + 1718 + 1817

Answer:

Unit’s place of 1211 = 8

Unit’s place of 1312 = 1

Unit’s place of 1718 = 9

Unit’s place of 1817 = 8

Sum of unit’s places = 8 + 1 + 9 + 8 = 26

Unit’s digit = 6

**Shortcut 81:**

Finding remainder

* R[(a*b*c)/p]=R(a/p)*R(b/p)*R(c/p)*

Question:

Find the remainder when 212 x 313 x 414 is divided by 5.

Answer:

Using the above concept we can find the remainder for each term and then we can multiply.

R(212/5) = 2

R(313/5) = 3

R(414/5) = 4

2 x 3 x 4 = 24

The value we got is still greater than 5, so again divide it and find the remainder.

R(24/5) = 4

The remainder of the expression is = 4