CAT Quant · Study & Practice

Divisibility Rules

AreaNumber System DifficultyEasy–Moderate CAT weightage1–2 direct questions + a constant tool inside remainders, factors, base systems and DI

Divisibility is the quiet workhorse of CAT Number System. A clean set of rules lets you decide whether one integer divides another without doing the long division — and that single skill feeds almost every other Number System topic: factors and the factor-count formula, remainders, HCF and LCM, last-digit and unit-digit problems, and the "find the missing digit" puzzles that XAT and SNAP love. The rules themselves are short (test 2 by the last digit, 3 and 9 by the digit sum, 11 by the alternating sum), but the real CAT skill is two layers deeper. First, breaking a composite divisor into coprime parts: a number is divisible by 12 exactly when it is divisible by 3 and by 4, because 3 and 4 share no common factor. Second, working backwards — given an unknown digit in a number, find every value that makes the whole thing divisible by 7, 8 or 11. This chapter covers the standard rules for 2 through 11, the coprime-factorisation method for composite divisors, the digit-sum family, and the reverse "unknown digit" technique, each with worked CAT-style examples, the fastest mental method, and the traps that cost marks.

Topics

1 Rules for 2 to 11 4 worked · 5 practice · 8-Q test Start → 2 Advanced Divisibility 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.

  • For any composite divisor, factor it into COPRIME parts and test each: 12 = 3×4, 15 = 3×5, 18 = 2×9, 72 = 8×9, 99 = 9×11.
  • Powers of 2: divisibility by 2^k depends only on the last k digits (4 → last 2, 8 → last 3, 16 → last 4).
  • Digital root: keep adding digits to one digit. Root 9 ⇒ ÷ 9; root divisible by 3 ⇒ ÷ 3.
  • By 11, use the alternating digit sum; a palindrome with an even number of digits is always divisible by 11.
  • Missing-digit problems: turn the rule into an equation in the unknown digit, then sweep 0–9 — rarely more than two answers.
  • By 7, 11 and 13 together: group the number in blocks of 3 from the right and alternate-subtract — that block test catches all three at once (since 7×11×13 = 1001).

⚠️ Common mistakes & traps

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

  • Splitting a composite divisor into NON-coprime factors (12 = 6×2, 72 = 6×12) — it gives false positives.
  • Checking divisibility by 6 with only the 3-test and forgetting the number must also be even.
  • Using the last TWO digits for 8 (it needs the last THREE) or the last one for 4 (it needs two).
  • In the 11-rule, taking a plain digit sum instead of the ALTERNATING sum, or mishandling the sign at the leftmost digit.
  • In reverse problems, stopping at the first valid digit when the rule (3, 9 or 11) allows two values in 0–9.

📈 CAT exam insight & PYQ analysis

In CAT, pure divisibility questions are rare as standalone items but the skill is everywhere — factor-count questions, remainder problems, last-two-digit and base-system sets all lean on it. XAT, SNAP and IIFT are more direct, regularly asking "find the digit that makes N divisible by 8 / 11 / 72" or "how many values of the missing digit work". The recurring high-value pattern is composite divisors handled through coprime factorisation, and the two-unknown-digit problem solved by combining two rules (commonly 8 and 9 for 72, or 9 and 11 for 99). Prioritise the 8, 9 and 11 rules and the coprime-split method — they unlock the most marks per minute.

🎴 Flashcards — instant recall

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

Divisibility test for 4?Tap to reveal
Last two digits form a multiple of 4
Divisibility test for 8?Tap to reveal
Last three digits form a multiple of 8
Divisibility test for 3 and 9?Tap to reveal
Digit sum divisible by 3 / by 9
Divisibility test for 11?Tap to reveal
Alternating digit sum is 0 or a multiple of 11
Divisibility test for 6?Tap to reveal
Divisible by 2 AND 3
Test 12 using which coprime factors?Tap to reveal
3 and 4
Test 72 using which coprime factors?Tap to reveal
8 and 9
Why does 12 = 6 × 2 fail as a test?Tap to reveal
6 and 2 are not coprime (e.g. 18 passes both, not ÷ 12)
Divisibility by 2^k depends on?Tap to reveal
The last k digits
For 67A2 to be ÷ 3, A can be?Tap to reveal
0, 3, 6 or 9
Block test for 7, 11 and 13 uses?Tap to reveal
Groups of 3 digits, alternating subtract (7×11×13 = 1001)
An even-length palindrome is always divisible by?Tap to reveal
11

📌 Quick revision

Test 2 and 5 by the last digit, 4 by the last two, 8 by the last three, and 2^k by the last k digits. Use the digit sum for 3 and 9 (digital root), and the ALTERNATING digit sum for 11. For 6 the number must be both even and divisible by 3. For any composite divisor, split it into COPRIME factors and test each — 12 = 3×4, 15 = 3×5, 72 = 8×9 — never use non-coprime parts like 6×2. For missing-digit problems, write the rule as an equation in the unknown and sweep 0–9; with two unknowns, combine two rules (e.g. 8 and 9 for 72). Watch the classic traps: wrong number of trailing digits, plain sum instead of alternating sum for 11, and forgetting that 6 needs evenness.

Chapter test

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

You have truly mastered Divisibility Rules 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