CAT Quant · Study & Practice

Cyclicity

AreaNumber System DifficultyEasy–Moderate CAT weightage1–2 questions (Number System + inside Remainders, Last-digit / units-digit problems)

Cyclicity is the study of how the unit digit (and more generally the remainder) of a number repeats in a fixed cycle as you raise it to higher and higher powers. Multiply 7 by itself again and again and the last digit marches through 7, 9, 3, 1, 7, 9, 3, 1… forever — a clean cycle of length 4. CAT loves this because a question like "find the unit digit of 7^203" looks intimidating but collapses to a 10-second mental step once you know the cycle. The whole topic rests on one observation: only the unit digit of the base controls the unit digit of any power, and that unit digit repeats with a period of 1, 2, or 4. So you never compute the huge number — you find where the exponent lands inside the cycle using its remainder when divided by the cycle length (usually mod 4). The same cyclic behaviour shows up in remainder problems, last-two-digit problems, and pattern questions on the GMAT, XAT and SNAP too. This chapter builds the two skills CAT actually tests: reading the unit digit straight off the power, and using power-mod-4 to locate the right term of the cycle quickly and without error.

Topics

1 Cycle of Powers 4 worked · 5 practice · 8-Q test Start → 2 Pattern Period 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 — 23^45 behaves exactly like 3^45.
  • Group the digits: {0,1,5,6} period 1, {4,9} period 2, {2,3,7,8} period 4. Memorise this split.
  • For period-4 digits, reduce the exponent mod 4; only the last two digits of the exponent affect mod 4 (100 is a multiple of 4).
  • Remainder 0 (exponent divisible by 4) means take the LAST term of the cycle, not the first.
  • For 4 and 9: odd power keeps the digit (4, 9); even power flips it (6, 1). Skip the full cycle.
  • For a product or sum, work with each part’s unit digit separately, then combine and take the last digit again.

⚠️ Common mistakes & traps

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

  • Treating remainder 0 (n divisible by 4) as the 1st term instead of the 4th/last term of the cycle.
  • Reducing the exponent mod 10 or mod the base instead of mod the cycle length (4 for 2,3,7,8).
  • Forgetting that 4 and 9 have period 2, and wrongly using a 4-cycle for them.
  • Using the whole base for mod-4 reduction — it is the EXPONENT that gets reduced, not the base.
  • Assuming every base cycles with period 4; 0,1,5,6 never change and 4,9 flip every other power.

📈 CAT exam insight & PYQ analysis

Cyclicity appears in CAT mainly as a quick units-digit or last-two-digits question, and as the engine behind tougher remainder problems in the Number System. The recurring patterns are: unit digit of a single large power (e.g. 7^203), unit digit of a product or sum of powers, and remainders of a^n by small numbers using the order/period. XAT and SNAP ask it more directly as a standalone last-digit problem. Difficulty is Easy–Moderate alone but climbs when fused with remainder theory or factorial-trailing-zero style work. Prioritise the {2,3,7,8} cycles and the power-mod-4 step — that single skill clears the majority of these questions in seconds.

🎴 Flashcards — instant recall

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

Which unit digits have period 1?Tap to reveal
0, 1, 5, 6 (the digit never changes)
Which unit digits have period 2?Tap to reveal
4 and 9
Which unit digits have period 4?Tap to reveal
2, 3, 7 and 8
Cycle of 2?Tap to reveal
2, 4, 8, 6
Cycle of 3?Tap to reveal
3, 9, 7, 1
Cycle of 7?Tap to reveal
7, 9, 3, 1
Cycle of 8?Tap to reveal
8, 4, 2, 6
Unit digit of 4^odd and 4^even?Tap to reveal
4 (odd), 6 (even)
For period-4 digits, what do you do with the exponent?Tap to reveal
Reduce it mod 4
Exponent divisible by 4 → which term of the cycle?Tap to reveal
The 4th (last) term
Unit digit of 7^203?Tap to reveal
3 (203 mod 4 = 3 → 3rd term of 7,9,3,1)
Why only the last two digits of the exponent matter for mod 4?Tap to reveal
Because 100 is divisible by 4

📌 Quick revision

Cyclicity says the unit digit of a power repeats in a fixed cycle, and only the base’s unit digit matters. Digits 0,1,5,6 have period 1; 4 and 9 have period 2; 2,3,7,8 have period 4 with cycles (2,4,8,6), (3,9,7,1), (7,9,3,1) and (8,4,2,6). To pick a unit digit, reduce the exponent mod 4: remainders 1,2,3 give the 1st,2nd,3rd term and remainder 0 gives the last term. Only the last two digits of the exponent affect mod 4. The same period idea solves remainder problems, since a^n mod m also cycles — find the period, reduce the exponent, read off the answer.

Chapter test

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

You have truly mastered Cyclicity 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