Number System and Divisibility

Overview

Number System and Divisibility appears about 1 times per paper across the last four AFCAT solved papers, placing it in the high yield band of Numerical Ability.

Number system is the most reliable scoring pocket inside AFCAT Numerical Ability. The paper carries 18 to 22 quant questions, and one of them is almost always built from divisibility rules, HCF and LCM, remainders or unit-digit cycles. The content is finite, the methods are mechanical, and the arithmetic is small. If you commit the divisibility table and the unit-digit cycles to memory and practise 40 to 50 mixed items, you should be able to lock this one mark in under a minute every time.

This page walks through every sub-area that has appeared in AFCAT-style papers — number classification, divisibility rules from 2 to 13, HCF and LCM with the product identity, the fraction formulas, modular arithmetic, cyclic unit digits, remainders of large powers, the factor-count and factor-sum formulas, successive division, and the three or four traps the setter likes to repeat. The worked examples at the end show the working you should actually put on paper.

Why number system gives reliable marks

Three reasons this topic is worth front-loading in your prep.

• The syllabus is closed. Unlike data interpretation or word problems, number system has a fixed list of rules and identities. You can finish the theory in two sittings.
• Arithmetic stays small. A typical AFCAT item asks for the remainder when a power is divided by 7, or the smallest four-digit multiple of 36. The numbers are tame; the trick is choosing the right method.
• The penalty maths is friendly. AFCAT marks each correct answer +3 and each wrong answer −1. A 75 percent accuracy on this topic still leaves you net positive. With cycle and rule recall, accuracy here often crosses 90 percent.

Practically, you should treat this as a one-mark guaranteed pick. Spend roughly 45 to 75 seconds on the item, attempt it confidently, and move on.

Types of numbers

Every quantity you meet in AFCAT lives inside one of the boxes below. The set inclusion runs from inside out — every natural number is also a whole number, every whole number is an integer, every integer is rational, every rational is real, and every real is complex.

Even, odd, prime, composite and coprime

• Even: Integer divisible by 2. Even × anything = even. Sum of two evens is even; sum of two odds is even; sum of one even and one odd is odd.
• Odd: Integer not divisible by 2. Odd × odd = odd. Square of an odd number is always odd.
• Prime: Natural number greater than 1 with exactly two distinct positive divisors, namely 1 and itself.
• Composite: Natural number greater than 1 that has at least one divisor other than 1 and itself.
• Coprime (relatively prime): Two numbers whose HCF is 1. They need not be prime individually; 8 and 15 are coprime.

Smallest prime: 2, which is also the only even prime. Every other prime is odd.

Why 1 is not prime: A prime must have exactly two distinct positive divisors. The number 1 has only one positive divisor (itself), so it fails the definition. Excluding 1 also keeps the fundamental theorem of arithmetic clean — every integer greater than 1 then has a unique prime factorisation.

Primes you should recognise instantly up to 100: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97. That is 25 primes. Knowing them on sight saves five to ten seconds on factorisation items.

Divisibility rules table

Memorise this entire table. Most AFCAT divisibility items are solved in seconds by applying one rule from here.

HCF and LCM — definitions and methods

• HCF (Highest Common Factor) of two or more numbers is the largest positive integer that divides each of them exactly.
• LCM (Least Common Multiple) is the smallest positive integer that is a multiple of each of them.

Three methods, pick whichever is fastest for the given numbers:

1. Prime factorisation: Write each number as a product of primes. HCF = product of the lowest powers of common primes. LCM = product of the highest powers of all primes appearing in any number.
2. Division (Euclidean) method: For HCF of two numbers, divide the larger by the smaller, then the divisor by the remainder, and keep going until the remainder is 0. The last non-zero divisor is the HCF.
3. Listing multiples or factors: Works only for small numbers; useful as a cross-check.

Example with prime factorisation. Take 72 and 120. Then 72 = 2³ × 3² and 120 = 2³ × 3 × 5. HCF = 2³ × 3 = 24. LCM = 2³ × 3² × 5 = 360.

Example with Euclidean method. HCF of 252 and 105. 252 = 2 × 105 + 42. 105 = 2 × 42 + 21. 42 = 2 × 21 + 0. HCF = 21.

HCF times LCM equals product (two numbers only)

For any two positive integers a and b:

HCF(a, b) × LCM(a, b) = a × b

This identity is the single most useful tool in HCF and LCM questions. Given any three of the four quantities — HCF, LCM, a, b — you can solve for the fourth without factorising.

A second useful fact: HCF always divides LCM. So if a question says the HCF is 12 and the LCM is 80, reject the data — 12 does not divide 80.

HCF and LCM of fractions

Reduce each fraction to lowest terms first, then apply:

• HCF of fractions = HCF of numerators ÷ LCM of denominators.
• LCM of fractions = LCM of numerators ÷ HCF of denominators.

Example. HCF and LCM of 2/3, 4/9 and 8/27. Numerators 2, 4, 8 → HCF 2, LCM 8. Denominators 3, 9, 27 → HCF 3, LCM 27. So HCF of the fractions = 2 / 27 and LCM of the fractions = 8 / 3.

Remainder theorem and modular arithmetic basics

The notation a mod n means the remainder when a is divided by n. So 17 mod 5 = 2, because 17 = 3 × 5 + 2.

The two identities that make remainder problems tractable:

• Sum rule: (a + b) mod n = ((a mod n) + (b mod n)) mod n.
• Product rule: (a × b) mod n = ((a mod n) × (b mod n)) mod n.

These let you reduce each block of a long expression modulo n before combining, which keeps the arithmetic small.

Example. Remainder when 47 × 53 is divided by 6. 47 mod 6 = 5; 53 mod 6 = 5. So (47 × 53) mod 6 = (5 × 5) mod 6 = 25 mod 6 = 1.

Negative remainders. If a calculation gives a negative value, add n to bring it into the range 0 to n minus 1. So −3 mod 7 = 4.

Cyclic remainders and unit-digit patterns

The unit digit of aⁿ depends only on the unit digit of a and the value of n. For each base digit there is a fixed cycle:

Method for unit digit of aⁿ:

1. Read the unit digit of a and look up its period p.
2. Compute n mod p. If the remainder is 0, take the last digit of the cycle; otherwise take the digit at position equal to the remainder.

Example. Unit digit of 1,283²⁰²⁶. Unit digit of base is 3, period 4. 2026 mod 4 = 2. The second entry of the cycle (3, 9, 7, 1) is 9. Answer: 9.

Remainders of large powers

For remainders when aⁿ is divided by m, the cycle method usually beats every other tool.

1. Compute a mod m. Call it r.
2. Compute r², r³, r⁴, ... mod m until you see a 1 or a repeat. Let the cycle length be k.
3. Reduce the exponent n modulo k; raise r to that reduced exponent and take mod m.

Example. Remainder when 3¹⁰⁰ is divided by 7. Powers of 3 mod 7: 3, 2, 6, 4, 5, 1, 3, 2, ... cycle length 6. 100 mod 6 = 4. The fourth entry is 4. Remainder = 4.

Fermat’s little theorem (light touch). If p is prime and a is not divisible by p, then a^{p − 1} ≡ 1 (mod p). This means powers of a mod p cycle within length p − 1. For AFCAT you rarely need to invoke the theorem by name — listing the first few powers and spotting the cycle is faster.

Number of factors and sum of factors

Write n in prime-factorised form as n = p^a × q^b × r^c, where p, q, r are distinct primes.

Example. Number of divisors of 360. 360 = 2³ × 3² × 5. Count = 4 × 3 × 2 = 24.

Example. Sum of divisors of 72. 72 = 2³ × 3². Sum = [(2⁴ − 1)/1] × [(3³ − 1)/2] = 15 × 13 = 195.

Successive division

In successive division, the quotient from one division becomes the dividend for the next. The remainders are read in the order they arise.

Example. A number N gives remainder 2 when divided by 5; the quotient gives remainder 3 when divided by 4; the next quotient gives remainder 1 when divided by 3. Find the smallest N.

Work backwards. Let the final quotient be 1 (the smallest possible). Then the previous quotient = 3 × 1 + 1 = 4. The original quotient = 4 × 4 + 3 = 19. N = 5 × 19 + 2 = 97.

Common AFCAT trap patterns

• Smallest n-digit number divisible by k. The smallest n-digit number is 10ⁿ⁻¹. Divide by k, take the ceiling, multiply back by k. Going the other way (largest n-digit) is similar but uses the floor.
• Two numbers from sum and difference of HCF or LCM. If sum is S and HCF is h, write the numbers as ha and hb with a and b coprime; then a + b = S/h. Enumerate coprime pairs.
• Remainder of a long concatenated number. Use the divisibility rule of the divisor block by block, not direct division. For 11, alternating digit sum still works on a 12-digit number.
• Largest number that divides several numbers leaving the same remainder r. Subtract r from each, then take the HCF. If remainders differ, subtract each from its own number first.
• Missing digit problems. If a four-digit number 3a2b is divisible by 9, then 3 + a + 2 + b must be a multiple of 9 — solve for a, b. Combine with divisibility by 4 (last two digits) when both rules are given.

Time budget

AFCAT gives you 120 minutes for 100 questions. That is 72 seconds per item on the global average, but easy topics should subsidise the hard ones.

• Divisibility rule item: 25 to 40 seconds.
• HCF or LCM with two-number identity: 30 to 45 seconds.
• Unit-digit of large power: 30 to 45 seconds.
• Remainder of aⁿ by m: 60 to 90 seconds (cycle detection takes time).
• Number-of-factors with factorisation: 45 to 75 seconds.

Target attempt rate on this topic: 100 percent of the items you see. If a remainder cycle is not appearing in 30 seconds, move on and come back — never spend more than 90 seconds on a one-mark item.

Worked AFCAT-style examples

Exam-day strategy

1. Memorise the divisibility table from 2 to 13 and 25. These are pure recall items that should never take more than 40 seconds.
2. Use HCF × LCM = product for any two-number question instead of factorising both numbers.
3. For remainders of large powers, always look for a short cycle first. Most useful moduli (4, 6, 7, 8, 9, 11) yield cycles of length 2 to 6.
4. Lock the unit-digit cycles: period 1 for digits 0, 1, 5, 6; period 2 for 4 and 9; period 4 for 2, 3, 7, 8. Then unit-digit questions reduce to one division by 4.
5. For successive division, always work backwards from the smallest possible final quotient.
6. If a question gives the same remainder when divided by several numbers, subtract that remainder from each and then take the HCF.
7. Aim for 45 to 75 seconds on this topic. Never spend more than 90 seconds on a one-mark item; mark and move on.

Frequently asked questions

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