CAT Quant · Study & Practice

Unit Digits

AreaNumber System DifficultyEasy–Moderate CAT weightage1–2 questions (directly + as a step inside remainder, factorial and number-property problems)

The unit digit — the very last digit of a number — is one of the highest return-on-effort ideas in CAT Number System. You will almost never be asked to actually compute 7^123; instead you are asked only for its last digit, and that collapses a terrifying-looking power into a ten-second mental step. The engine behind it is cyclicity: the unit digits of successive powers of any number repeat in a short cycle (length 1, 2 or 4), so the whole problem reduces to finding where your exponent lands inside that cycle. Once the last digit is tamed, the natural next question is the last two digits, which CAT and XAT use to separate fast solvers from the rest. This chapter builds both skills cleanly: the cyclicity table for every digit 0–9, how to handle the last digit of products and sums without multiplying the whole thing, and the last-two-digit toolkit — powers of numbers ending in 1, the all-important rule that even numbers ride the 76 family while 5-enders settle into 25 or 76 patterns, and the binomial shortcut for bases close to a multiple of 100. Each topic comes with worked CAT-style examples, the fastest exam method, and the traps that cost careless students an easy mark.

Topics

1 Last Digit 4 worked · 5 practice · 8-Q test Start → 2 Last Two Digits 4 worked · 5 practice · 8-Q test Start →

⚡ CAT shortcuts & speed methods

The fastest ways to crack this chapter under time pressure — the techniques that separate a 95+ percentiler from the rest.

  • Strip the base to its unit digit first: 2467^85 behaves exactly like 7^85.
  • Only digits 2,3,7,8 need full cyclicity (length 4) — reduce the exponent mod 4; 0,1,5,6 stay fixed, 4 and 9 just alternate.
  • Remainder 0 (mod 4) means take the LAST term of the cycle, never the base itself.
  • For last two digits of a number ending in 1: tens = (tens-of-base × exponent) mod 10, unit = 1.
  • 76 and 25 are automorphic for last two digits — their powers always end in 76 and 25 respectively.
  • Anchor every even base on 2^10 = …24, then use 24^even = …76 and 24^odd = …24 to collapse huge powers.

⚠️ Common mistakes & traps

CAT is designed so that careless errors here cost you marks. Internalise each trap before the exam.

  • Treating exponent ≡ 0 (mod 4) as the 0th term and writing the base digit instead of the 4th cycle term.
  • Reducing the exponent modulo 10 instead of modulo the cycle length (4).
  • For last two digits of base-1 numbers, forgetting to take (tens × exponent) modulo 10 and writing the full product.
  • Assuming 24^n always ends in 24 — it alternates 24 (odd) and 76 (even).
  • Including 5! and beyond when only the unit digit of a factorial sum is needed — every factorial from 5! ends in 0.

📈 CAT exam insight & PYQ analysis

In CAT and XAT, pure "find the unit digit" questions have become rarer as standalone items, but the skill is constantly reused inside remainder, factorial, and number-property problems where the final answer hinges on the last one or two digits. The recurring patterns are: unit digit of a large power or a product of powers, unit digit of a factorial sum (always 3 from the 1!+2!+3!+4! tail), and last-two-digit questions on bases ending in 1, 25 or 76, or bases near 100 solved by the binomial trick. Difficulty is Easy–Moderate, so these are marks you should never lose; speed and a memorised cyclicity table decide the percentile.

🎴 Flashcards — instant recall

Tap a card to reveal the answer. Drill these until they are automatic.

Which unit digits have cycle length 1?Tap to reveal
0, 1, 5, 6
Cycle of unit digit 2?Tap to reveal
2, 4, 8, 6 (length 4)
Cycle of unit digit 3?Tap to reveal
3, 9, 7, 1 (length 4)
Cycle of unit digit 7?Tap to reveal
7, 9, 3, 1 (length 4)
Cycle of unit digit 8?Tap to reveal
8, 4, 2, 6 (length 4)
Exponent ≡ 0 (mod 4): which cycle term?Tap to reveal
The 4th (last) term
Unit digit of 1!+2!+…+100!?Tap to reveal
3 (only 1+2+6+24 = 33 counts)
Last two digits of 76^n?Tap to reveal
76 for every n ≥ 1
Last two digits of 25^n?Tap to reveal
25 for every n ≥ 1
Last two digits rule for base ending in 1?Tap to reveal
tens = (tens×exp) mod 10, unit = 1
Last two digits of 2^10?Tap to reveal
24
24^even and 24^odd end in?Tap to reveal
even → 76, odd → 24

📌 Quick revision

The unit digit of a power depends only on the base’s unit digit and the exponent mod 4. Memorise the cycles: 0,1,5,6 fixed; 4,9 alternate; 2(2,4,8,6), 3(3,9,7,1), 7(7,9,3,1), 8(8,4,2,6). Remainder 0 means the 4th term, not the base. For products multiply unit digits; for factorial sums only 1!+2!+3!+4! = 33 matters (unit digit 3). Last two digits: base ending in 1 → tens = (tens×exp) mod 10 with unit 1; 76 and 25 are automorphic; anchor even bases on 2^10 = 24 and use 24^even = 76, 24^odd = 24; bases near 100 fall to the binomial (100 ± a)^n ≡ (±a)^n (mod 100).

Chapter test

📝 Unit Digits — Chapter Test
25 fresh questions · timed · full solutions
Start chapter test →
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🏆 Vidaara CAT success checklist

You have truly mastered Unit Digits when you can tick every box below.

  • Recall every formula in this chapter without looking them up
  • Solve each topic’s practice set with at least 80% accuracy
  • Use the chapter shortcuts to cut your solving time in half
  • Spot and avoid every common trap listed above
  • Score 80%+ on the timed chapter test

📋 Chapter mastery scorecard

Track where you stand. Aim for the target before moving to the next chapter.

Skill checkpointTarget
Concept theory & formulas understood100%
Topic practice sets attempted (2 topics)2/2
Best topic-test score— → 80%+
Chapter test score— → 80%+
Flashcards drilled to instant recall12 cards