Number System and Number Series quietly decide your AFCAT Numerical Ability score. They look small, but they feed almost every other chapter — divisibility, remainders, factors, HCF, LCM and series patterns appear again and again. Master the classification of numbers, divisibility rules and series logic here, and you will solve these questions in under a minute, banking easy marks while others burn precious time.
Why this topic is your easiest scorer
In every AFCAT paper the Numerical Ability section carries a reliable handful of direct questions on number properties and number series. These are not lengthy word problems — they are short, formula-driven and fully solvable in your head once you know the underlying rules. Because they reward memory and pattern recognition rather than heavy calculation, they are the closest thing to free marks the section offers.
The smart strategy is simple: lock down divisibility rules, factor logic and series patterns so well that you never have to do long division or guesswork. A single series question can take a careless student two full minutes; a trained one finishes it in twenty seconds. Across a paper, that saved time is exactly what lets you reach the longer arithmetic questions with a calm mind.
Number System feeds HCF, LCM, remainders, simplification, ratio and even Profit & Loss. Strong basics here lift your entire AFCAT Numerical Ability score, not just one chapter.
Types of numbers you must know
Every number in the exam belongs to one of these families. Knowing the boundary cases is exactly where marks are won and lost, because setters love to test the edges rather than the obvious middle.
- Natural numbers (N): counting numbers 1, 2, 3, … (start at 1).
- Whole numbers (W): natural numbers plus 0, i.e. 0, 1, 2, 3, …
- Integers (Z): …, −2, −1, 0, 1, 2, … (negatives, zero and positives).
- Rational numbers: any number written as p/q where q ≠ 0, e.g. 3/4, −5, 0.25. Their decimal form either terminates or repeats.
- Irrational numbers: non-terminating, non-repeating decimals like √2, √3 and π.
- Real numbers: all rationals and irrationals together — every number on the number line.
- Prime numbers: exactly two factors, 1 and itself (2, 3, 5, 7, 11…). 2 is the only even prime.
- Composite numbers: more than two factors (4, 6, 8, 9…).
- Co-prime numbers: two numbers whose HCF is 1, such as 8 and 15, even though neither is prime on its own.
1 is neither prime nor composite. 0 is a whole number and an integer but not a natural number, and 0 is even. Keep these boundary facts ready — they are tested almost every year.
Even, odd and prime quick facts
These tiny facts save you on tricky options where two answers look plausible.
- Even × any number = even; Odd × odd = odd.
- Even ± even = even; Odd ± odd = even; Even ± odd = odd.
- Sum of two odd numbers is always even; sum of an even count of odd numbers is even, an odd count is odd.
- The product of any two consecutive integers is always even, because one of them must be even.
- A number is prime if it is not divisible by any prime less than or equal to its square root.
To check if 97 is prime, test only primes up to √97 ≈ 9.8, i.e. 2, 3, 5, 7. None divide 97, so 97 is prime. You never need to test beyond the square root — this alone saves huge time on prime-checking questions.
Divisibility rules that save time
Memorise these — they turn long division into a one-glance check, and they reappear inside HCF, LCM and remainder questions too.
- By 2: last digit is even (0, 2, 4, 6, 8).
- By 3: sum of digits divisible by 3.
- By 4: last two digits form a number divisible by 4.
- By 5: last digit is 0 or 5.
- By 6: divisible by both 2 and 3.
- By 7: double the last digit, subtract from the rest; if the result is divisible by 7, so is the number.
- By 8: last three digits divisible by 8.
- By 9: sum of digits divisible by 9.
- By 10: last digit is 0.
- By 11: difference between the sum of digits at odd places and even places is 0 or a multiple of 11.
Worked check for 11: in 825, odd-place digits (from the right) = 5 + 8 = 13, even-place digit = 2. Difference = 13 − 2 = 11, which is a multiple of 11, so 825 is divisible by 11 (825 = 11 × 75). Worked check for 7: for 343, double the last digit (3 × 2 = 6), subtract from 34 to get 28, and since 28 is divisible by 7, so is 343 (343 = 73).
Factors, HCF and LCM links
Almost every number-property question rests on prime factorisation, so make it second nature. Write any number as a product of primes first, and the rest follows.
If N = ap × bq × cr in prime factors, then:
- Number of factors = (p+1)(q+1)(r+1).
- Sum of factors uses the product of (ap+1−1)/(a−1) terms for each prime.
- HCF = product of the lowest powers of common primes.
- LCM = product of the highest powers of all primes that appear.
- For two numbers, HCF × LCM = product of the two numbers. Use this when three of the four quantities are known.
- HCF is always a factor of the LCM; the LCM is always a multiple of each number.
Find the number of factors of 360 and its HCF and LCM with 84.
So 360 has 24 factors, the HCF is 12 and the LCM is 2520. Check: HCF × LCM = 12 × 2520 = 30240 = 360 × 84.
Finding the unit digit fast
Questions like “find the unit digit of 783” are pure shortcut play. The unit digit of successive powers repeats in a fixed cycle, so you never compute the full number.
- 2: 2, 4, 8, 6 (cycle of 4)
- 3: 3, 9, 7, 1 (cycle of 4)
- 7: 7, 9, 3, 1 (cycle of 4)
- 8: 8, 4, 2, 6 (cycle of 4)
- 4: 4, 6 (cycle of 2); 9: 9, 1 (cycle of 2)
- 0, 1, 5, 6: always end in the same digit, whatever the power
Divide the power by the cycle length (usually 4) and use the remainder to pick the digit. A remainder of 0 means take the last digit of the cycle.
For 783: cycle of 4, and 83 ÷ 4 leaves remainder 3. The 3rd digit in 7's cycle (7, 9, 3, 1) is 3, so the unit digit is 3. The same trick handles unit digits of products: the unit digit of 23 × 47 is simply the unit digit of 3 × 7 = 21, which is 1.
Remainders and the division rule
The division rule is the backbone of every remainder question, and AFCAT loves wrapping it inside a short word problem.
Dividend = (Divisor × Quotient) + Remainder, where 0 ≤ Remainder < Divisor. The remainder can never equal or exceed the divisor.
- When a number is divided by 9, the remainder equals the remainder of its digit sum divided by 9 — a fast checking trick.
- To find the remainder of a large power, reduce the base modulo the divisor first, then apply cyclicity.
- If a number leaves remainder r on division by d, then adding or subtracting any multiple of d keeps the same remainder.
What is the remainder when 250 is divided by 7?
So 250 leaves a remainder of 4 when divided by 7. Spotting the short cycle is far faster than any long calculation.
Types of number series in AFCAT
A number series hides one rule that links each term to the next. Train your eye to scan for these patterns in a fixed order so you never waste time:
- Difference (arithmetic): constant difference, e.g. 3, 7, 11, 15… (add 4).
- Product (geometric): constant ratio, e.g. 2, 6, 18, 54… (×3).
- Square based: 1, 4, 9, 16, 25… (n²), sometimes with a shift like n²+1.
- Cube based: 1, 8, 27, 64… (n³), or n³ − 1, etc.
- Alternating: two patterns woven together on odd and even positions.
- Mixed: a combination such as ×2 then +1, applied repeatedly.
- Difference of differences: the gaps themselves form a series, e.g. 2, 5, 10, 17… (differences 3, 5, 7…).
- Prime or special: 2, 3, 5, 7, 11… (primes), or factorials 1, 2, 6, 24…
Always check the difference between consecutive terms first. If that fails, check the ratio, then squares and cubes, then alternate terms. Following a fixed order stops you from staring blankly.
A 4-step shortcut for any series
Use this routine and you will rarely get stuck on a series question.
- Write the differences between consecutive terms.
- If the differences are equal → arithmetic. If they grow steadily, take the differences again (second-level differences).
- If differences do not settle, test the ratio by dividing each term by the previous one.
- Still stuck? Split into alternate terms, or check for squares, cubes, primes and factorials.
Do not force the first pattern you see. In 1, 4, 9, 16, 25 the differences (3, 5, 7, 9) tempt you, but the real rule is simply n² — always confirm a candidate rule against the next given term before committing.
More solved number-series patterns
Here are several worked patterns so you recognise them instantly in the exam.
Find the next number: 3, 6, 11, 18, 27, ?
So the answer is 38. The terms also equal n² + 2 (1+2, 4+2, 9+2…, 36+2 = 38), confirming it.
Find the missing term: 2, 5, 10, 17, 26, ?
The answer is 37. The differences 3, 5, 7, 9 also confirm the next gap is 11, giving 26 + 11 = 37.
Find the next number: 1, 4, 2, 8, 3, 12, ?
The answer is 4. Two simple series are interleaved — splitting them apart makes the rule obvious.
Find the missing term: 5, 11, 23, 47, ?
The answer is 95. When neither a clean difference nor ratio fits, test a combined operation like double-plus-one.
Common traps to avoid
AFCAT setters love these slips, and avoiding them protects easy marks:
- Treating 1 as prime, or wrongly calling 2 “not prime” because it is even.
- Forgetting that the remainder must always be smaller than the divisor.
- Mixing up HCF (largest common factor) with LCM (smallest common multiple).
- Using the wrong cyclicity — check whether the unit-digit cycle is 2 or 4.
- Counting factors wrongly by forgetting to add 1 to each prime power before multiplying.
- Stopping at the first pattern in a series instead of verifying with a later term.
When two answer options both seem to fit a series, pick the rule that fits every given term, not just the first two. A rule that explains all terms is almost always the intended one.
Previous-year style practice
Q. What is the unit digit of 17256?
Answer: The unit digit depends only on 7. 7 has a cycle of 4 (7, 9, 3, 1). 256 ÷ 4 leaves remainder 0, so we take the last digit of the cycle, which is 1. The unit digit is 1.
Q. The HCF of two numbers is 12 and their LCM is 144. If one number is 36, find the other.
Answer: HCF × LCM = product of the numbers, so 12 × 144 = 36 × other. The other number = (12 × 144) / 36 = 1728 / 36 = 48.
Q. Find the missing term: 5, 11, 23, 47, ?
Answer: Each term is double the previous plus 1 (5×2+1=11, 11×2+1=23, 23×2+1=47). Next = 47×2+1 = 95.
Quick revision
- 1 is neither prime nor composite; 2 is the only even prime; 0 is even and a whole number.
- Divisibility: by 3 and 9 use digit sums; by 11 use the alternating digit difference; by 4 and 8 check the last two or three digits.
- Factors of apbq = (p+1)(q+1); HCF uses lowest prime powers, LCM uses highest, and HCF × LCM = product of two numbers.
- Unit digits repeat in cycles — divide the power by the cycle length and read off the remainder.
- For any series: check differences first, then ratio, then squares/cubes/alternates/mixed rules.
- Confirm a pattern against every given term before choosing an option.
Practise five mixed series and ten divisibility or factor checks daily for two weeks — speed on this chapter quietly lifts your entire AFCAT Numerical Ability score.
Frequently asked questions
How many questions on number system and series come in AFCAT?
AFCAT does not fix a count, but the Numerical Ability section usually carries a few direct questions on number properties, divisibility, factors and number series. They are short, high-accuracy questions, so treat them as near-guaranteed marks and never leave them for the end.
What is the fastest way to find the unit digit of a large power?
Use cyclicity. Most digits repeat in a cycle of 4 (a few, like 4 and 9, repeat in a cycle of 2). Divide the power by the cycle length and use the remainder to pick the unit digit; a remainder of 0 means you take the last digit of the cycle.
Is 1 a prime number?
No. A prime number has exactly two distinct factors. Since 1 has only one factor (itself), it is neither prime nor composite. This is one of the most common AFCAT traps, so keep it firmly in mind.
How do I count the factors of a number quickly?
Write the number in prime factorised form, such as 360 = 2^3 x 3^2 x 5. Add 1 to each power and multiply: (3+1)(2+1)(1+1) = 24 factors. This avoids listing factors one by one and almost never goes wrong.
How do I decide which series pattern to test first?
Always check the difference between consecutive terms first. If that fails, test the ratio, then look for squares, cubes, primes, alternating sub-series or mixed operations like double-plus-one. Verify your rule against every given term before answering.
What is the relationship between HCF and LCM?
For any two numbers, HCF multiplied by LCM equals the product of the two numbers. This lets you find one quantity quickly when the other three are known, and it is a frequent shortcut in AFCAT problems.
Related AFCAT Numerical Ability topics
Want a teacher to walk you through AFCAT Numerical Ability?
Cavalier's AFCAT batches break every topic into classroom sessions with daily practice, tests and doubt-clearing.