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Aptitude Guide

30 topics · Numerical, Reasoning, Verbal, Advanced

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Numerical

Number System — Unit Digits, Divisibility & HCF/LCM

Unit digits follow repeating cycles when you raise a number to a power — for example, 7¹ ends in 7, 7² ends in 9, 7³ ends in 3, 7⁴ ends in 1, then it repeats. Divisibility rules let you check if a number divides cleanly without doing long division. HCF is the largest number that divides both, LCM is the smallest number both divide into — and HCF × LCM always equals the product of the two numbers.

Key FormulasMemorise these before attempting problems
1Cyclicity lengths: {0,1,5,6} -> 1, {4,9} -> 2, {2,3,7,8} -> 4
2Unit digit of a^b = Unit digit of [ (last digit of a) ^ (b mod cyclicity) ]
3If b mod cyclicity == 0, use the cyclicity length as the effective exponent.
4HCF(a, b) * LCM(a, b) = a * b
5If d = k * q (k is a factor of d), then N mod k = (N mod d) mod k

Method — Step by Step

  1. 1Unit digit of aᵇ depends ONLY on the last digit of a. Find its cyclicity.
  2. 2Cyclicity: 0,1,5,6 → cycle 1. 4,9 → cycle 2. 2,3,7,8 → cycle 4.
  3. 3Compute b mod (cycle length). If result = 0, take the LAST element of the cycle.
  4. 4HCF × LCM = Product of the two numbers.
  5. 5Remainder shortcut: if N = d×q + r and d is a multiple of k, then N mod k = r mod k.

Worked Example

Find the unit digit of 7⁹⁵.

1Cycle of 7: (7, 9, 3, 1) — length 4.
295 ÷ 4 = 23 remainder 3.
3Take the 3rd element of the cycle = 3.
4Unit digit of 7⁹⁵ is 3.

Practice Problems

Problem 1 of 8Easy

Find the unit digit of 4¹⁷.