Practice Questions on Simplification and Shortcuts

'BODMAS' Rule:

This rule depicts the correct sequence in which the operations are to be executed, so as to find out the value of given expression.

Here B - Bracket,
O - of,
D - Division,
M - Multiplication,
A - Addition and
S - Subtraction

Thus, in simplifying an expression, first of all the brackets must be removed, strictly in the order (), {} and ||.

After removing the brackets, we must use the following operations strictly in the order:
(i) of (ii) Division (iii) Multiplication (iv) Addition (v) Subtraction.

Test of Divisibility:

By 2 - A number is divisible by 2, if the units place is 0 or divisible by 2.
By 3 - A number is divisible by 3, when the sum of digits of given number is divisible by 3.
By 4 - A number is divisible by 4, if the last two digits of the number are 0 or divisible by 4.
By 5 - A number is divisible by 5, if the units place is 0 or 5.
By 6 - A number is divisible by 6, if it is both divisible by 2 & 3 ( use divisibility test by 2 & 3)
By 7 - to check divisibility by 7, use 5 as a check multiplier.We multiply the unit digit with check multiplier
& then we add this answer to the number after removing the units digit.If the answer is evenly divisible by 7
then the number is evenly divisible by 7.
By 8 - A number is divisible by 8, if the last three digits are 0 or divisible by 8.
By 9 - A number is divisible by 9, when the sum of digits of given number is divisible by 9.
By 11 - A number is divisible by 11, if the difference between sum of odd digits & sum of even digits is divisble by 11.
By 12 - A number is divisible by 12, if it is both divisible by 3 & 4
By 13 - Method is similar to that of 7. Only that the check multiplier in this case is 4
By 17 - Method is similar to that of 7 or 13. Only that the check multiplier in this case is 12
By 19 - Method is similar to that of 7 or 13 or 17. Only that the check multiplier in this case is 2
By 25 - A number is divisible by 25, if the last two digits of the number are 0 or divisible by 25.
By 75 - A number is divisible by 75, if it is divisible by 3 and 25.
By 125 - A number is divisible by 125, if the last three digits of the number are 0 or divisible by 125.

If p is a prime number,then for any whole number W, (W^p - W) is divisible by p.
Eg: 8³ -8 = 512 -8 = 504. 504 is divisible by 3.

xⁿ + yⁿ = (x+y)(x^n-1 -x^n-2 *y +......+ y^n-1 ),when n is odd. Therefore when n is odd, xⁿ + yⁿ is divisble by (x+y)

xⁿ - yⁿ = (x+y)(x^n-1 -x^n-2*y +......+ y^n-1 ),when n is even. Therefore when n is even, xⁿ - yⁿ is divisble by (x+y)

xⁿ - yⁿ = (x-y)(x^n-1 -x^n-2*y +......+ y^n-1 ),even when n is both odd or even. Therefore when n is odd or even, xⁿ - yⁿ is divisble by (x-y)