Q1 — NUMERICAL: EVERY QUESTION TYPE

What it looks like · the move · a worked example

This is a recognition catalogue, not a textbook. The other files teach method. This one trains you to look at a question and know which shape it is within five seconds — which is the whole game when you have ~90 seconds each and cannot finish the section.

Format for every type: Looks like → The move → Example.

1 · NUMBER SYSTEM — 175 questions

1A · Remainders — 38% (the biggest sub-type in the exam)

Type 1 — Remainder of a big product

Looks like: "Find the remainder when 123 × 124 × 125 is divided by 7." The move: Replace each number by its own remainder first, then multiply. Example: 123→4, 124→5, 125→6. (4×5×6)=120, 120 mod 7 = 1.

Type 2 — Remainder of a large power

Looks like: "Remainder when 2¹⁰⁰ is divided by 7." The move: List powers until they cycle. Take the exponent mod the cycle length. Example: 2¹=2, 2²=4, 2³=1 → cycle 3. 100 mod 3 = 1 → 2.

If the exponent divides evenly, use the last element of the cycle, not the first.

Type 3 — Base is one less than the divisor

Looks like: "Remainder when 6¹⁰⁰ ÷ 7" or "15⁵⁰ ÷ 16" The move: Write it as (−1)ⁿ. Even power → 1. Odd power → divisor − 1. Example: 6 ≡ −1 (mod 7) → (−1)¹⁰⁰ = 1. Instant, no working.

Type 4 — Prime divisor, huge power (Fermat)

Looks like: "Remainder when 3¹⁰⁰ ÷ 7" The move: If p is prime, a^(p−1) ≡ 1. So the cycle length is p−1. Example: 3⁶ ≡ 1 (mod 7). 100 mod 6 = 4 → 3⁴ = 81 ≡ 4.

Type 5 — Composite divisor (Euler)

Looks like: "Last two digits of 7¹⁰⁰" = "7¹⁰⁰ mod 100" The move: φ(n) gives the cycle. φ(100)=40. Reduce the exponent mod 40. Example: 100 mod 40 = 20 → 7²⁰. Since 7⁴ = 2401 ≡ 1, 7²⁰ = (7⁴)⁵ ≡ 1.

Type 6 — Remainder of a factorial sum

Looks like: "Remainder when 1!+2!+…+100! is divided by 5." The move: From 5! onward every term contains 5, so all vanish. Only sum the first few. Example: 1+2+6+24 = 33, 33 mod 5 = 3.

Type 7 — Same remainder from several numbers

Looks like: "Largest number dividing 62, 132, 237 leaving the same remainder." The move: Take differences — the remainder cancels. Then HCF. Example: 70, 105, 175 → HCF = 35.

Type 8 — Number leaving a fixed remainder

Looks like: "Smallest number leaving remainder 3 when divided by 5, 6 and 8." The move: LCM + remainder. Example: LCM(5,6,8)=120 → 123.

Variant: if divisor − remainder is the same each time (4 by 5, 5 by 6, 7 by 8 → gap 1), the answer is LCM − gap = 119.

1B · Two-digit / digit manipulation — 11%

Type 9 — Digits reversed

Looks like: "A two-digit number exceeds its reverse by 27. Find it." The move: Write it as 10a + b. Difference from reverse is always 9(a−b). Example: 9(a−b)=27 → a−b=3 → 41, 52, 63, 74, 85, 96 all work; extra condition picks one.

Type 10 — Number vs sum/square of its digits

Looks like: "A two-digit number is 18 less than the square of the sum of its digits." The move: Set 10a+b, form the equation, test digit sums 4–9. Small search space. Example: (10a+b)+18 = (a+b)² → two such numbers exist.

Read whether it is "square of the sum" or "sum of the squares" — TCS asks both.

1C · The quick-win formulas — 5% each

Type 11 — Unit digit of a power

Looks like: "Unit digit of 7¹⁰³." The move: Take the exponent mod 4, read the digit's cycle. Example: 7 cycles 7,9,3,1. 103 mod 4 = 3 → third element → 3.

Type 12 — Trailing zeros in a factorial

Looks like: "How many zeros at the end of 100!?" The move: Count the 5s: ⌊n/5⌋+⌊n/25⌋+⌊n/125⌋… Example: 20 + 4 = 24.

Type 13 — Number of factors

Looks like: "How many factors does 72 have?" The move: Prime factorise, add 1 to each power, multiply. Example: 72 = 2³·3² → (3+1)(2+1) = 12. (Odd factors: drop the 2s → 3.)

Type 14 — HCF / LCM word problem

Looks like: "Largest tile that fits a floor" → HCF. "Bells ring together again" → LCM. The move: Identify which word signals which. Then HCF × LCM = product of the numbers.

Type 15 — Recursive function

Looks like: "f(0)=0, f(2n)=f(n), f(2n+1)=f(n)+1. Find f(2018)." The move: Unroll downward — do not look for a formula. Example: Answer 6 — it counts the 1s in binary (2018 = 11111100010).

2 · TIME, SPEED & DISTANCE — 140 questions

Before anything: convert km/h → m/s by × 5/18. Every time.

2A · Trains — 16%

Type 16 — Train crosses a pole/man

Looks like: "A 150 m train at 72 km/h crosses a pole. Time?" The move: Distance = train length only. Example: 72 km/h = 20 m/s → 150/20 = 7.5 s.