Number Series: Spotting the Pattern

Some of the most satisfying questions on the whole Civil Service Exam are number-series questions: you stare at a row of numbers like "2, 5, 8, 11, ___" and suddenly the hidden rule clicks and the answer appears. This chapter teaches you to make that click happen on demand — through a systematic checklist rather than lucky guessing — and it also covers the number-theory facts (factors, multiples, primes, divisibility) that quietly power many exam items. By the end you'll have a reliable method for both.

1. Lesson Overview

This topic joins two related skills. Number series questions show a sequence with a missing term and ask you to find the pattern and fill the gap. Number theory questions test facts about how whole numbers behave: factors, multiples, prime numbers, greatest common factor (GCF), least common multiple (LCM), and divisibility rules.

On the Civil Service Exam, number-series items are a staple of Numerical Reasoning — you can count on several — and number-theory facts appear both directly and inside other problems (simplifying fractions needs GCF; finding "when events coincide" needs LCM). Together they touch 5 to 9 items on a typical exam.

The difficulty ranges from easy to challenging. The easy series follow one simple rule; the hard ones interleave two patterns or hide a "pattern of the differences." The secret is not raw cleverness but a fixed order of things to check — the same checklist every time — so you never freeze in front of a sequence again.

2. Learning Objectives

After completing this lesson you will be able to:

• Apply a step-by-step checklist to identify the rule behind any number series.
• Recognize the common series types: constant difference, constant ratio, changing difference, alternating, and squares/cubes/primes/Fibonacci.
• Verify a suspected rule against more than one gap before committing.
• Define and find factors, multiples, primes, GCF, and LCM.
• Use divisibility rules to test numbers quickly without long division.
• Solve real exam problems that use GCF (grouping, tiling) and LCM (coinciding events).
• Avoid the guess-once-and-give-up habit that causes most misses.

3. Prerequisites

You should be comfortable with:

• Basic operations, including squares (7² = 49) and cubes (2³ = 8).
• Multiplication tables up to at least 12 — fast recall speeds up pattern spotting enormously.
• Simple mental subtraction (to compute the differences between terms).

A quick refresher on squares and cubes, because they hide inside many series. Squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 (that's 1² through 10²). Cubes: 1, 8, 27, 64, 125 (1³ through 5³). Memorizing these two short lists lets you instantly recognize a "squares series" or "cubes series" that would otherwise stump you.

4. Why This Topic Matters

Pattern recognition and number sense are everywhere:

• Everyday planning. "The MRT arrives every 4 minutes; the bus every 6 — when do they arrive together?" is an LCM question.
• Sharing and grouping. "What's the largest identical group I can make from 24 boys and 36 girls?" is a GCF question.
• Money and schedules. Payments every 15 days, a bonus every 30 — figuring out when both land together uses LCM.
• Construction. "What's the biggest square tile that fits a 240 cm by 360 cm floor exactly?" is GCF again.
• Coding and logic. Spotting sequences underlies spreadsheets, programming, and data work.

Beyond the exam, this topic builds "number sense" — a feel for how numbers relate — which quietly makes all other math faster and more accurate.

5. Core Concepts

We'll handle series first, then number theory.

The number-series checklist

When you see a sequence, run through these in order. The first one that fits is almost always the answer.

1. Constant difference (arithmetic). Each term adds (or subtracts) the same number.

2, 5, 8, 11, ___ → add 3 each time → 14.

2. Constant ratio (geometric). Each term multiplies (or divides) by the same number.

3, 6, 12, 24, ___ → multiply by 2 → 48.

3. Changing difference. The differences themselves follow a pattern.

1, 2, 4, 7, 11, ___ → differences are 1, 2, 3, 4, so add 5 → 16.

4. Alternating / interleaved. Two separate series woven together — often the odd positions follow one rule and the even positions another.

1, 10, 2, 20, 3, ___ → odd positions 1, 2, 3…; even positions 10, 20, …; next is an even position → 30.

5. Squares, cubes, primes, or Fibonacci.

1, 4, 9, 16, ___ → perfect squares → 25. 1, 8, 27, 64, ___ → perfect cubes → 125. 2, 3, 5, 7, 11, ___ → prime numbers → 13. 1, 1, 2, 3, 5, 8, ___ → each term is the sum of the two before it (Fibonacci) → 13.

6. Multiply-and-add rules. Each term is transformed by "×k then ± c."

1, 3, 7, 15, 31, ___ → each term is ×2 then +1 → 63.

The working method

Underneath the sequence, write the differences between consecutive terms. If those differences are constant, you have type 1. If the differences grow in a pattern, type 3. If differences don't help, try ratios (divide each term by the previous) for type 2. If neither works, suspect an alternating series and look at every other term. Finally, compare against squares, cubes, and primes.

Golden rule: once you think you've found the rule, test it on at least two gaps, not one. A rule that fits a single gap might be a coincidence; a rule that fits the whole sequence is the answer.

Number theory — the vocabulary

• A factor (or divisor) of a number divides into it evenly, with no remainder. Factors of 12: 1, 2, 3, 4, 6, 12.
• A multiple of a number is that number times any whole number. Multiples of 4: 4, 8, 12, 16, 20, …
• A prime number has exactly two factors: 1 and itself. Examples: 2, 3, 5, 7, 11, 13. The number 2 is the only even prime. The number 1 is neither prime nor composite (it has just one factor).
• A composite number has more than two factors (e.g., 12).
• Prime factorization expresses a number as a product of primes: 60 = 2 × 2 × 3 × 5 = 2² × 3 × 5.

Analogy: Primes are the "atoms" of numbers — you can't break them into smaller whole-number factors, and every other number is built by multiplying primes together, just as molecules are built from atoms.

GCF and LCM — opposite tools

• The Greatest Common Factor (GCF) is the largest number that divides evenly into two or more numbers. GCF(12, 18) = 6. Use it to split into equal groups or simplify fractions.
• The Least Common Multiple (LCM) is the smallest number that two or more numbers all divide into. LCM(12, 18) = 36. Use it to find when repeating events coincide or a common denominator.

Fast method using prime factorization: 12 = 2² × 3 and 18 = 2 × 3².

• GCF = multiply the lowest power of each shared prime: 2¹ × 3¹ = 6.
• LCM = multiply the highest power of every prime that appears: 2² × 3² = 36.

Memory tip: GCF is the Greatest thing that fits into both (so it's small-ish); LCM is the Least thing both fit into (so it's big-ish). GCF ≤ each number ≤ LCM.

Position-based and factorial series

Some series are best understood by their position n (1st term, 2nd term, …). A term might equal n², or n² + n, or n³. Recognizing "this is just the position squared" cracks a series that resists differences.

2, 6, 12, 20, 30, ___ → these are n² + n (1+1, 4+2, 9+3, 16+4, 25+5) → next is 36 + 6 = 42.

A special fast-growing type is the factorial-style series, where each term multiplies by the next counting number:

1, 2, 6, 24, 120, ___ → ×2, ×3, ×4, ×5 → next is ×6 → 720.

When a series explodes upward far faster than doubling, suspect a growing multiplier like this.

Remainders and "divides leaving a remainder"

A useful number-theory idea for the exam: if a number leaves a remainder when divided, subtract the remainder to get a number that divides evenly. This turns a remainder problem into a plain GCF problem.

What is the largest number that divides both 62 and 92, leaving a remainder of 2 in each case? Subtract the remainder: it must divide 62 − 2 = 60 and 92 − 2 = 90 exactly. So the answer is GCF(60, 90) = 30.

Divisibility rules

These give a fast yes/no without dividing:

• 2: last digit is even (0, 2, 4, 6, 8).
• 3: the digits add up to a multiple of 3.
• 4: the last two digits form a number divisible by 4.
• 5: last digit is 0 or 5.
• 6: divisible by both 2 and 3.
• 8: the last three digits form a number divisible by 8.
• 9: the digits add up to a multiple of 9.
• 10: last digit is 0.

Example: Is 3,471 divisible by 3? Digit sum 3 + 4 + 7 + 1 = 15, which is divisible by 3, so yes. Divisible by 9? 15 is not a multiple of 9, so no.

With the toolkit assembled, let's picture it, list it, and drill it.

6. Visual Learning Suggestions

• [Illustration Suggestion] Differences underneath. Write a series in a row, then draw little arrows between terms with the difference labeled on each arrow. Seeing "+3, +3, +3" or "+1, +2, +3" makes the pattern jump out.
• [Illustration Suggestion] Two-color interleave. For an alternating series, color the odd-position terms blue and the even-position terms red, then read each color as its own simple series.
• [Illustration Suggestion] Factor tree. Draw 60 branching into 6 × 10, then 6 → 2 × 3 and 10 → 2 × 5, ending in the primes 2, 2, 3, 5. This shows prime factorization concretely.
• [Illustration Suggestion] GCF vs. LCM number line. Mark the multiples of 4 and 6 on a line; the first place they overlap (12) is the LCM. Separately, list factors of 12 and 18; the biggest shared one (6) is the GCF.

7. Formula Library

The handy identity: for any two numbers, GCF × LCM = the product of the numbers. If you know three of these four, you can find the fourth. For 12 and 18: 6 × 36 = 216 = 12 × 18. ✓

8. Pattern Recognition

For number theory: "largest… equal groups / biggest tile / simplify" → GCF; "smallest… again together / common denominator / next time both" → LCM.

9. Problem-Solving Framework

Step 1 — For a series, write the differences between consecutive terms. ↓ Step 2 — Check the checklist in order: constant difference → constant ratio → changing difference → alternating → squares/cubes/primes/Fibonacci → ×k ± c. ↓ Step 3 — State the rule in words ("multiply by 2 then add 1"). ↓ Step 4 — Test it on two or more existing gaps to be sure. ↓ Step 5 — Apply it to the missing term and select the answer.

For number-theory items: decide whether the situation wants GCF (splitting/grouping) or LCM (coinciding/repeating), then use prime factorization to compute it.

10. Worked Examples

Beginner

Example 1. Find the next term: 7, 14, 21, 28, ___. Solution: Constant difference of 7 → 35. Difficulty: ★☆☆☆☆

Example 2. Find the next term: 1, 3, 9, 27, ___. Solution: Constant ratio ×3 → 81. Difficulty: ★☆☆☆☆

Intermediate

Example 3 (changing difference). Find the next term: 2, 3, 6, 11, 18, ___. Thinking: Differences are 1, 3, 5, 7 (odd numbers) → next difference is 9. Solution: 18 + 9 = 27. Check: the differences 1, 3, 5, 7, 9 are consistent. ✓ Difficulty: ★★☆☆☆

Example 4 (alternating). Find the next term: 5, 20, 6, 19, 7, ___. Thinking: Odd positions: 5, 6, 7 (add 1). Even positions: 20, 19, … (subtract 1). The blank is an even position → 19 − 1. Solution: 18. Difficulty: ★★★☆☆

Example 5 (squares nearby). Find the next term: 2, 5, 10, 17, 26, ___. Thinking: These are 1²+1, 2²+1, 3²+1, 4²+1, 5²+1 → next is 6² + 1. Solution: 36 + 1 = 37. Alternative: differences are 3, 5, 7, 9 → next 11 → 26 + 11 = 37. Same answer, two routes. ✓ Difficulty: ★★★☆☆

Advanced

Example 6 (multiply-and-add). Find the next term: 3, 7, 15, 31, 63, ___. Thinking: Each term is ×2 then +1: 3→7→15→31→63. Solution: 63 × 2 + 1 = 127. Difficulty: ★★★☆☆

Example 7 (GCF word problem). A florist has 24 roses and 36 tulips and wants to make identical bouquets using all the flowers with none left over. What is the greatest number of bouquets? Thinking: "Identical groups, greatest number, nothing left" → GCF. Solution: 24 = 2³ × 3; 36 = 2² × 3². GCF = 2² × 3 = 12 bouquets (each with 2 roses and 3 tulips). Difficulty: ★★★☆☆

Civil Service Exam Level

Example 8 (LCM word problem). At a terminal, one jeepney leaves every 8 minutes and another every 12 minutes. If both leave together at 6:00 a.m., when will they next leave together? Thinking: "Next time both together" → LCM. Solution: 8 = 2³, 12 = 2² × 3. LCM = 2³ × 3 = 24 minutes. Next together at 6:24 a.m. Difficulty: ★★★★☆

Example 9 (divisibility reasoning). Which of these is divisible by both 3 and 4: 132, 146, 150, 168? Thinking: Divisible by 4 → last two digits divisible by 4; by 3 → digit sum divisible by 3. Solution: 168: last two digits 68 ÷ 4 = 17 ✓, digit sum 1 + 6 + 8 = 15 divisible by 3 ✓ → 168. (132 is divisible by 4 and by 3 too — 32÷4=8, 1+3+2=6 — so check the intended single answer carefully; both 132 and 168 qualify, which is exactly the kind of "more than one fits" trap to watch. If only one choice may be picked, the item is flawed; on a real exam, pick the one the key intends, but knowing why both work is the mastery.) Difficulty: ★★★★☆

Example 10 (odd-one-out). Which number does not belong: 121, 144, 169, 180, 196? Thinking: The others are perfect squares (11², 12², 13², 14²). 180 is not. Solution: 180 is the odd one out. Check: 121 = 11², 144 = 12², 169 = 13², 196 = 14². ✓ Difficulty: ★★★☆☆

Example 11 (decreasing changing difference). Find the next term: 100, 96, 88, 76, 60, ___. Thinking: Differences are −4, −8, −12, −16 (going down by 4 more each time) → next is −20. Solution: 60 − 20 = 40. Difficulty: ★★★☆☆

Example 12 (factorial-style). Find the next term: 1, 2, 6, 24, 120, ___. Thinking: Each term multiplies by the next counting number (×2, ×3, ×4, ×5). Solution: 120 × 6 = 720. Difficulty: ★★★★☆

Example 13 (Fibonacci-style). Find the next term: 2, 3, 5, 8, 13, ___. Thinking: Each term is the sum of the two before it. Solution: 8 + 13 = 21. Difficulty: ★★★☆☆

Example 14 (LCM word problem). What is the smallest number of candies that can be shared equally among 6, 8, or 12 children with none left over? Thinking: "Smallest number that 6, 8, and 12 all divide into" → LCM. Solution: 6 = 2·3, 8 = 2³, 12 = 2²·3 → LCM = 2³ · 3 = 24 candies. Difficulty: ★★★★☆

11. Exam Tricks

• Try ratios when adding fails. If differences are wild and growing fast, the rule is probably multiplication, not addition. Divide consecutive terms to reveal it.
• The interleave disguise. A series that zig-zags (up, down, up, down) is usually two simple series taped together. Split it before giving up.
• "Almost squares." Series like 2, 5, 10, 17 look mysterious until you notice they're squares plus a constant. Keep the squares list handy.
• The one-gap coincidence. A rule that fits only the last gap is a trap. Always verify against an earlier gap too.
• GCF vs. LCM wording. "Greatest… into" is GCF; "least… both go into / next together" is LCM. Underline the keyword; picking the wrong one flips the whole answer.

12. Common Mistakes

• Testing only one pattern type (usually differences) and quitting instead of trying ratios or interleaving.
• Confirming a rule on a single gap without checking a second.
• Missing an alternating series and forcing a single continuous rule.
• Confusing GCF and LCM — GCF is the largest shared factor (small); LCM is the smallest shared multiple (large).
• Forgetting the number facts: 1 is not prime, and 2 is the only even prime.
• Divisibility slips: using the "sum of digits" rule (which is for 3 and 9) on the wrong divisor.

13. Shortcuts

• Differences-then-ratios is the fastest triage: write differences first; if they're not constant or patterned, switch to ratios immediately.
• GCF × LCM = product lets you find one from the other: knowing GCF(12,18) = 6 and 12 × 18 = 216 gives LCM = 216 ÷ 6 = 36 instantly.
• Prime-factorize once to get both GCF (lowest powers) and LCM (highest powers) from the same work.
• Digit-sum test settles divisibility by 3 and 9 in seconds — no division needed.
• Recognize the short lists: squares to 100 and cubes to 125 should be instant recall; many "hard" series are just these in disguise.

14. Memory Techniques

• "Differences first, ratios second, split third, squares last." The checklist order in five words.
• "Test two gaps." Never trust a one-gap rule.
• "GCF fits IN, LCM they fit INTO." Settles the two forever; GCF is small, LCM is large.
• "2 is the lonely even prime; 1 is neither." The two facts examiners love to test.
• "3 and 9 love digit sums." The divisibility shortcut for those two.

15. Real Civil Service Exam Strategy

• Time: an easy series should take under 20 seconds; a tricky one up to a minute. If a series resists the whole checklist for over a minute, mark it and move on.
• Always write the differences — even a quick scribble makes the pattern visible far faster than staring.
• Don't force elegance. The rule can be a plain "+7." Simple answers are common; don't overthink toward something fancy.
• For GCF/LCM, name the tool first. Decide "this is a GCF question" (grouping) or "an LCM question" (coinciding) before computing.
• Use divisibility rules to eliminate. On "which is divisible by…" items, the rules let you cross off wrong choices in seconds.

16. Practice Questions

Easy

1. Next term: 4, 8, 12, 16, ___.
2. Next term: 2, 4, 8, 16, ___.
3. Is 234 divisible by 3?

Medium

4. Next term: 3, 4, 6, 9, 13, ___.
5. Next term: 1, 4, 9, 16, 25, ___.
6. Find GCF and LCM of 8 and 12.

Hard

7. Next term: 2, 6, 12, 20, 30, ___.
8. Next term: 5, 11, 23, 47, ___ (each term ×2 then +1).

Challenge

9. Two lighthouses flash every 15 seconds and every 18 seconds, together now. After how many seconds do they flash together again?
10. Which is the odd one out: 8, 27, 64, 100, 125?
11. Next term: 1, 2, 4, 8, 16, ___ — then explain why the answer is not necessarily unique.
12. What is the largest number that divides 43 and 91 leaving a remainder of 1 in each case?

Answers and Explanations

1. 20. Constant difference 4.
2. 32. Constant ratio ×2.
3. Yes. Digit sum 2 + 3 + 4 = 9, divisible by 3.
4. 18. Differences 1, 2, 3, 4 → next 5 → 13 + 5 = 18.
5. 36. Perfect squares (6²).
6. GCF = 4, LCM = 24. 8 = 2³, 12 = 2²·3 → GCF 2² = 4, LCM 2³·3 = 24. (Check: 4 × 24 = 96 = 8 × 12. ✓)
7. 42. Differences 4, 6, 8, 10 → next 12 → 30 + 12 = 42. (Also: these are n² + n → 6²+6 = 42.)
8. 95. 47 × 2 + 1 = 95.
9. 90 seconds. LCM of 15 and 18: 15 = 3·5, 18 = 2·3² → LCM = 2·3²·5 = 90.
10. 100. The others are perfect cubes (2³, 3³, 4³, 5³); 100 is a square, not a cube.
11. 32 by the simplest rule (×2). The teaching point: a rule that fits the given terms may not be the only one — the sequence 1, 2, 4, 8, 16 also famously continues 31 under a different (regions-of-a-circle) rule. On the exam, choose the simplest rule that fits all shown terms; here that's doubling, giving 32.
12. 6. Subtract the remainder: the number must divide 43 − 1 = 42 and 91 − 1 = 90 exactly → GCF(42, 90) = 6. Check: 43 ÷ 6 = 7 r1 and 91 ÷ 6 = 15 r1. ✓

17. Summary

For number series, don't guess — run the checklist: write the differences first; if not constant, look for a pattern in the differences, then ratios, then an interleaved pair, then squares/cubes/primes/Fibonacci, and finally a ×k ± c rule. Always verify your rule on two gaps before answering. For number theory, know the vocabulary (factor, multiple, prime, composite), remember that 1 is not prime and 2 is the only even prime, and use prime factorization to get both GCF (lowest shared powers — for grouping) and LCM (highest powers — for coinciding events). Lean on divisibility rules to test numbers instantly. Systematic checking, not cleverness, is what turns this topic into easy points.

18. Cheat Sheet

19. Frequently Asked Questions

I found a rule but I'm not sure it's the intended one. What do I do? Test it on every gap in the sequence. The correct rule fits all of them; a coincidental rule breaks somewhere. If two different rules both fit every term and give different next values, the item is ambiguous — pick the simpler rule.

Is 1 a prime number? No. A prime must have exactly two distinct factors; 1 has only one factor (itself). And 1 is not composite either — it's in a category of its own.

How do I find LCM quickly for the exam? Prime-factorize both numbers and take the highest power of every prime that appears. Or, if you already have the GCF, use LCM = (product of the numbers) ÷ GCF.

When is it GCF and when is it LCM? GCF for "split into the largest equal groups / biggest tile / simplify a fraction." LCM for "when do repeating events happen together again / smallest common denominator."

Do I need to memorize primes? Knowing the primes up to about 30 (2, 3, 5, 7, 11, 13, 17, 19, 23, 29) is very helpful and quick to memorize. Beyond that, use divisibility rules to test.

20. Mastery Checklist

• ☐ I write the differences under a series before anything else.
• ☐ I check patterns in a fixed order and don't quit after one try.
• ☐ I recognize alternating/interleaved series.
• ☐ I know the squares to 100 and cubes to 125 by sight.
• ☐ I verify a suspected rule on two or more gaps.
• ☐ I can prime-factorize a number.
• ☐ I can find GCF and LCM and know which one a problem needs.
• ☐ I remember 1 is not prime and 2 is the only even prime.
• ☐ I can apply the divisibility rules for 2, 3, 4, 5, 6, 8, 9, and 10.

Tick them all and both number series and number theory become dependable, quick points — plus you've sharpened the number sense that speeds up every other numerical topic.