### Chapter: Numbers

Important formulae and tricks and shortcuts of Numbers:

We are providing you Important Concepts and Tricks on Numbers which are usually asked in almost all competitive exams like IBPS, Govt. Exams and CAT and in quantitative aptitude of MNCs. Use these below given tricks to solve questions within minimum time. These tricks will be very helpful for your upcoming Competitive Exam.

1. Number Sets

a. Counting Numbers (Natural numbers)

1, 2, 3 ...

b. Whole Numbers

0, 1, 2, 3 ...

c. Integers

-3, -2, -1, 0, 1, 2, 3 ...

d. Rational Numbers

Rational numbers can be expressed as ab/ab where a and b are integers and b≠0b≠0
Examples: 112/112, 42/42, 00, −8/11−8/11 etc.

All integers, fractions and terminating or recurring decimals are rational numbers.

Formulas of Number Series

1. 1 + 2 + 3 + 4 + 5 + … + n = n(n + 1)/2
2. (12 + 22 + 32 + ..... + n2) = n ( n + 1 ) (2n + 1) / 6
3. (13 + 23 + 33 + ..... + n3) = (n(n + 1)/ 2)2
4. Sum of first n odd numbers = n2
5. Sum of first n even numbers = n (n + 1)

Mathematical Formulas

• (a + b)(a - b) = (a2 - b2)
• (a + b)2 = (a2 + b2 + 2ab)
• (a - b)2 = (a2 + b2 - 2ab)
• (a + b + c)2 = a2 + b2 + c2 + 2(ab + bc + ca)
• (a3 + b3) = (a + b)(a2 - ab + b2)
• (a3 - b3) = (a - b)(a2 + ab + b2)
• (a3 + b3 + c3 - 3abc) = (a + b + c)(a2 + b2 + c2 - ab - bc - ac)
• When a + b + c = 0, then a3 + b3 + c3 = 3abc
• (a + b)n = an + (nC1)an-1b + (nC2)an-2b2 + … + (nCn-1)abn-1 + bn

Shortcuts for number divisibility check

• A number is divisible by 2, if its unit's digit is any of 0, 2, 4, 6, 8.
• A number is divisible by 3, if the sum of its digits is divisible by 3.
• A number is divisible by 4, if the number formed by the last two digits is divisible by 4.
• A number is divisible by 5, if its unit's digit is either 0 or 5.
• A number is divisible by 6, if it is divisible by both 2 and 3.
• A number is divisible by 8, if the number formed by the last three digits of the given number is divisible by 8.
• A number is divisible by 9, if the sum of its digits is divisible by 9.
• A number is divisible by 10, if it ends with 0.
• A number is divisible by 11, if the difference of the sum of its digits at odd places and the sum of its digits at even places, is either 0 or a number divisible by 11.
• A number is divisible by 12, if it is divisible by both 4 and 3.
• A number is divisible by 14, if it is divisible by 2 as well as 7.
• Two numbers are said to be co-primes if their H.C.F. is 1. To find if a number, say y is divisible by x, find m and n such that m * n = x and m and n are co-prime numbers. If y is divisible by both m and n then it is divisible by x.

For solved problems on above formulas please visit below sections: