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Inequalities Cheatsheet
Every CAT inequalities formula in one place — linear to modulus, the wavy-curve method, AM-GM, Cauchy-Schwarz and Titu's Lemma — each with a worked example and the traps to avoid.
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Number System
CAT'26 QUANT CHEATSHEET
Every number-system formula and CAT shortcut you need for CAT 2026 — on one page.
Number System is the backbone of CAT quant — it appears directly and props up half of arithmetic and algebra besides. The syllabus is wide, but the marks cluster around a handful of ideas: how factors behave, how remainders collapse under Fermat and Euler, and how factorials hide powers of primes. This sheet distils the highest-yield of those into one page, from divisibility and factor counts through HCF-LCM, unit-digit cycles, modular arithmetic and base systems, each with a worked example in real numbers. It covers the CAT quant section’s most reusable tools — the ones examiners return to year after year. Keep it open while you practise, and after a mock check where you stand on the CAT score predictor to see which idea is costing you marks.
Dev note: the canonical route
/cheatsheets/optima-learn-number-system-cheatsheet is not live yet
(no /cheatsheets hub in the current sitemap). Built production-ready as a drop-in once it ships.Number System: every formula you need
1Classification of Numbers
How every real number is grouped and nested.
N ⊂ W ⊂ Z ⊂ Q ⊂ R | Q ∪ irrationals = R
Example: 0.75 = 3/4 is rational; 0.333… = 1/3 is rational; √2 = 1.414… never repeats, so irrational.
CAT Insight: A terminating or repeating decimal is always rational; only a non-repeating, non-terminating decimal is irrational.
2Divisibility Rules
Quick tests for the small divisors.
2: even · 4: last 2 digits · 8: last 3 digits
3, 9: digit sum · 11: alternating digit sum
3, 9: digit sum · 11: alternating digit sum
Example: 5148: digit sum = 18 (÷9 ✓); alternating sum 8−4+1−5 = 0 (÷11 ✓) → divisible by 9 and 11.
CAT Hack: For 6, 12 or 15, test the coprime factors separately: ÷6 means ÷2 and ÷3.
3Number of Factors
Count divisors from the prime factorisation.
N = pa · qb → factors = (a+1)(b+1)
Example: 72 = 23·32 → (3+1)(2+1) = 4·3 = 12 factors.
CAT Favourite: This factor-count setup recurs across CAT papers — always prime-factorise first.
4Sum of Factors
Add all divisors with a geometric-series product.
Sum = ∏ (pa+1 − 1)/(p − 1)
Example: 72 = 23·32: (24−1)/1 · (33−1)/2 = 15·13 = 195.
5Product of Factors
Multiply every divisor together at once.
Product = Nd/2, d = number of factors
Example: 12 = 22·3 → 6 factors → product = 12(6/2) = 123 = 1728.
CAT Insight: When d is odd (a perfect square), Nd/2 still works — the square root pairs with itself.
6HCF & LCM
The bridge between two numbers and their extremes.
HCF × LCM = product of the two numbers
Example: 12 and 18: HCF = 6, LCM = 36 → 6·36 = 216 = 12·18. ✓
CAT Hack: HCF of fractions = HCF(numerators)/LCM(denominators); for the LCM of fractions, swap them.
7Unit Digit (Cyclicity)
Last digits repeat in a short, fixed cycle.
unit digit of an: cycle length divides 4 → use n mod 4
Example: 7n cycles 7, 9, 3, 1. For 7102, 102 mod 4 = 2 → unit digit = 9.
Common Mistake: When n mod 4 = 0, take the 4th (last) term of the cycle, not the first.
8Remainder Basics
Remainders distribute over addition and multiplication.
(a · b) mod n = [(a mod n)(b mod n)] mod n
Example: (17·23) mod 5 = (2·3) mod 5 = 6 mod 5 = 1.
9Fermat's Little Theorem
A prime modulus collapses a high power.
ap−1 ≡ 1 (mod p), p prime, gcd(a, p) = 1
Example: 3100 mod 7: 36 ≡ 1, 100 = 6·16+4 → 34 = 81 ≡ 4 (mod 7).
CAT Favourite: Prime modulus with a large exponent — Fermat almost always cracks it.
10Euler's Totient Theorem
Fermat's rule generalised to any modulus.
aφ(n) ≡ 1 (mod n); φ(n) = n · ∏(1 − 1/p)
Example: φ(10) = 10(1−1/2)(1−1/5) = 4 → 34 = 81 ≡ 1 (mod 10).
CAT Insight: Use Euler when the modulus is not prime; φ(n) counts the integers coprime to n.
11Highest Power of a Prime in n!
Sum the floor-divisions by rising powers of p.
Ep(n!) = ⌊n/p⌋ + ⌊n/p2⌋ + ⌊n/p3⌋ + …
Example: power of 3 in 20!: ⌊20/3⌋ + ⌊20/9⌋ = 6 + 2 = 8.
12Trailing Zeros in n!
Count the fives — they are the bottleneck.
zeros in n! = ⌊n/5⌋ + ⌊n/25⌋ + ⌊n/125⌋ + …
Example: zeros in 100!: ⌊100/5⌋ + ⌊100/25⌋ = 20 + 4 = 24.
CAT Hack: There are always more 2s than 5s in n!, so the 5s alone decide the zero count.
13Base System Conversion
Positional value of digits in any base b.
(dk…d1d0)b = Σ di · bi
Example: (1011)2 = 1·8 + 0·4 + 1·2 + 1·1 = 11 in base 10.
CAT Insight: Decimal to base b: divide repeatedly by b and read the remainders bottom-up.
14Sum of Series
Closed forms for the three common sums.
Σn = n(n+1)/2
Σn2 = n(n+1)(2n+1)/6 · Σn3 = (Σn)2
Σn2 = n(n+1)(2n+1)/6 · Σn3 = (Σn)2
Example: 1 to 10: Σn = 55, Σn2 = 385, Σn3 = 552 = 3025.
CAT Favourite: Σn3 = (Σn)2 — the sum of cubes is the square of the sum.
15Perfect Squares
Perfect squares have an odd number of factors.
N is a perfect square ⇔ N has an odd number of factors
Example: 36 = 22·32 → (2+1)(2+1) = 9 factors (odd) → perfect square.
CAT Insight: Perfect squares can only end in 0, 1, 4, 5, 6 or 9 — never 2, 3, 7 or 8.
CAT exam shortcuts, traps & revision
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CAT Exam Shortcuts
- Factors of N = paqb: (a+1)(b+1); product = Nd/2
- HCF × LCM = product of the two numbers
- Unit digit: reduce the exponent mod 4 on the base's cycle
- Prime modulus: Fermat ap−1 ≡ 1; otherwise Euler aφ(n) ≡ 1
- Power of p in n!: Σ⌊n/pk⌋; trailing zeros count the 5s
- Σn = n(n+1)/2, Σn2 = n(n+1)(2n+1)/6, Σn3 = (Σn)2
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Most Common CAT Traps
- Cyclicity: n mod 4 = 0 means the 4th term of the cycle, not the first.
- Counting 2s instead of 5s when finding trailing zeros.
- Forgetting the higher prime powers in Legendre's formula (⌊n/p2⌋, …).
- Using Fermat when gcd(a, p) ≠ 1, or when the modulus is not prime.
- Treating a non-terminating repeating decimal as irrational — it is rational.
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30-Second Revision Box
- N ⊂ W ⊂ Z ⊂ Q ⊂ R; a repeating decimal is rational
- Factors: (a+1)(b+1)…; sum = ∏(pa+1−1)/(p−1); product = Nd/2
- HCF × LCM = product; unit digit via n mod 4
- Remainders: Fermat (prime), Euler (any n), (ab) mod n distributes
- n!: power of p = Σ⌊n/pk⌋; zeros = count of 5s
- Base b: Σdibi; perfect square ⇔ odd number of factors
Number System rewards recognition over raw computation — the moment you see “find the remainder” or “how many factors,” the right tool is usually one line away. Drill this sheet until factorising, cycling unit digits and applying Fermat are reflexes, then test them on full sets and track progress with the CAT score predictor. It pairs closely with the Binomial Theorem cheat sheet, since remainders and last-digit tricks overlap. For more guides, browse the Optima Learn blog or explore every study guide, work through the CAT exam hub, and when you want mentor-led prep, book a free CAT 2026 call.
