Ocegik / Number System
Number System Formulas for CAT Preparation
Revise number system formulas from Ocegik’s formula library, with named identities, expressions and usage notes grouped by topic.
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Number System: formulas by topic
Sums of Counting NumbersNumber System
Sum of first n natural numbersn(n + 1)2
Sum of first n odd natural numbersn2
Sum of first n even natural numbersn(n + 1)
Sum of squares of first n natural numbersn(n + 1)(2n + 1)6
Sum of cubes of first n natural numbers[n(n + 1)2]2
LCM & HCFNumber System
LCM–HCF product ruleL × H = a × bfor two numbers a, b with LCM L and HCF H
LCM of fractionsLCM of numeratorsHCF of denominators
HCF of fractionsHCF of numeratorsLCM of denominatorsfractions in lowest form
Least number leaving remainder r(LCM of x, y, z) + r
Series of such numbers(LCM of x, y, z) × n + r
Factors & DivisorsNumber System
1 is not a prime number
Number of factors(m + 1)(n + 1)(p + 1)…N = aᵐ × bⁿ × cᵖ × … with a, b, c distinct primes
Ways as product of 2 factors (N not a perfect square)12{(p+1)(q+1)(r+1)…}p, q, r are the powers of N's prime factors
Ways as product of 2 different factors (N a perfect square)12{(p+1)(q+1)(r+1)… − 1}
Ways as product of 2 factors (N a perfect square)12{(p+1)(q+1)(r+1)… + 1}
Sum of the factors of N[ap+1 − 1a − 1][bq+1 − 1b − 1][cr+1 − 1c − 1]…N = aᵖ × bᵍ × cʳ × …
Totient function φ(N)N(1 − 1/a)(1 − 1/b)(1 − 1/c)…φ(N) = count of numbers ≤ N that are coprime to N; a, b, c are N's prime factors
Euler's theoremPφ(N) mod N = 1P is any natural number coprime to N
Sum of numbers less than & coprime to Nφ(N) × N2
Ways to write N as product of 2 coprime numbers2n − 1n = number of distinct prime factors of N
Product of all factors of NN(number of factors)/2
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