Some quantitative questions are not calculations at all. They ask whether a number is divisible by 9, what digit a large power ends in, how many multiples of 7 sit between 100 and 500, or what the smallest number is that both 84 and 126 divide into evenly. All four are answerable in under thirty seconds by rules, and in three minutes by brute force — which is the same as not answerable at all under a 72-second clock.
Number properties also power everything downstream: simplifying a ratio, comparing two fractions, and cancelling inside an algebra fraction all depend on seeing factors quickly.
Three ideas cover almost every question in this family.
The model is incomplete in one practical way: prime factorisation is the slow method. The fast method is the digit rules in the next unit, and the two must be used together.
| Divisor | Test | Example |
|---|---|---|
| 2 | Last digit is even | 4,718 is even, 4,717 is not |
| 3 | Digit sum is a multiple of 3 | 4,812: 4+8+1+2 = 15, so yes |
| 4 | Last two digits form a multiple of 4 | 7,116: 16 is a multiple of 4, so yes |
| 5 | Ends in 0 or 5 | 2,445 ends in 5 |
| 6 | Divisible by 2 and by 3 | 4,812 is even and digit-sum 15, so yes |
| 8 | Last three digits form a multiple of 8 | 9,024: 024 is 24, so yes |
| 9 | Digit sum is a multiple of 9 | 4,815: 4+8+1+5 = 18, so yes |
| 10 | Ends in 0 | 13,470 |
| 11 | Alternating digit sum is a multiple of 11 (including 0) | 4,732: (2+7) − (3+4) = 2, so no |
Primes below 50: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47. Two facts people lose marks on: 1 is not prime, and 2 is the only even prime.
Squares worth knowing cold: 11² = 121, 12² = 144, 13² = 169, 14² = 196, 15² = 225, 16² = 256, 17² = 289, 18² = 324, 19² = 361, 20² = 400, 21² = 441, 22² = 484, 23² = 529, 24² = 576, 25² = 625. Cubes: 3³ = 27, 4³ = 64, 5³ = 125, 6³ = 216, 7³ = 343, 8³ = 512, 9³ = 729, 10³ = 1000, 11³ = 1331, 12³ = 1728.
Rational vs irrational. Rational = p/q for integers p, q with q ≠ 0. Every integer, fraction, terminating decimal (0.25 = 1/4) and repeating decimal (0.333... = 1/3) is rational. Irrational = cannot be written that way: root 2, root 3, root 5, pi. A square root is rational only when the radicand is a perfect square — root 16 = 4 is rational, root 8 = 2 root 2 is not (and simplifying never changes that). Negative signs do not matter: −root 16 = −4 is an integer, hence rational; the question −root 16 is an integer (not natural, not whole — those exclude negatives).
HCF (Highest Common Factor, also called GCD) is the largest number that divides both numbers exactly. LCM (Least Common Multiple) is the smallest number that both numbers divide into exactly. Read the wording first — it decides which one you need:
HCF of 12 and 18. Factors of 12: 1, 2, 3, 4, 6, 12. Factors of 18: 1, 2, 3, 6, 9, 18. The largest number on both lists is 6.
LCM of 4 and 6. Multiples of 4: 4, 8, 12, 16, ... Multiples of 6: 6, 12, 18, ... The first number on both lists is 12. Listing is also the fastest way to check an answer you got another way.
Step 1: factor both numbers into primes.
84 = 2^2 x 3 x 7 126 = 2 x 3^2 x 7
Step 2: the HCF takes every prime that appears in both numbers, each to its lowest power. The LCM takes every prime that appears in either number, each to its highest power.
HCF = 2^1 x 3^1 x 7^1 = 42 LCM = 2^2 x 3^2 x 7^1 = 252
Step 3 (two numbers only): check with the identity. HCF × LCM = 42 × 252 = 10,584 and 84 × 126 = 10,584. They match, so the answer is consistent.
Write both numbers in a row. Divide both by the smallest prime that divides both, and keep dividing the quotients the same way until the two results share no common factor. Then:
84 126 2 | 42 63 3 | 14 21 7 | 2 3 HCF = 2 x 3 x 7 = 42 LCM = 42 x 2 x 3 = 252
The last row (2 and 3) shares no factor, so you stop. The ladder produces both answers from one piece of work — no separate factor lists needed.
Digit sum: 4+8+3+5 = 20. 20 is not a multiple of 9, so no. Note that the same sum tells you the remainder when divided by 9 is 2 — useful immediately.
The cycle for 7 is 7, 9, 3, 1, length 4. Divide the exponent by 4: 43 leaves remainder 3, so take the third element of the cycle: 3. (Check with a small case: 7³ = 343, which ends in 3.)
Whole part of 500 ÷ 7 is 71 (7 × 71 = 497). Whole part of 99 ÷ 7 is 14 (7 × 14 = 98). 71 − 14 = 57.
This asks for the LCM — the smallest time that is a multiple of both intervals. From the ladder above: LCM = 42 × 2 × 3 = 252 seconds, which is 4 minutes 12 seconds. The distractor set will include 10,584 (the product) and 42 (the HCF).
Which of these numbers is divisible by 9?
What is the units digit of 235?
The HCF of two numbers is 6 and their LCM is 60. What is their product?
A number leaves remainder 3 when divided by 7. What remainder does its square leave when divided by 7?
What is the HCF of 24 and 36?
What is the LCM of 8 and 12?
What is the greatest number that divides 24, 36 and 48 exactly?
Which of these is irrational?
Without looking back: state the divisibility test for 9, and the identity that connects HCF, LCM and the two numbers.
The answer: the digit sum must be a multiple of 9; and HCF × LCM equals the product of the two numbers.
A number is divisible by 9 exactly when its is a multiple of 9. For any two positive integers, HCF × LCM equals their — but this identity fails for three or more numbers. The HCF of 84 and 126 is and their LCM is . The number 1 is prime, and the only even prime is . A "greatest number that divides" question asks for the ; a "smallest number divisible by" question asks for the . A number that can be written as p/q for integers is , while root(2) is .
A question asks: "What is the smallest number that is divisible by each of 12, 15 and 18?" Work out the answer and name the distractor you expect beside it.
Factorise: 12 = 2² × 3, 15 = 3 × 5, 18 = 2 × 3². The LCM takes the largest exponent of each prime: 2² × 3² × 5 = 180.
The expected distractors: 3,240 (the product, from forgetting that the numbers share factors), 3 (the common factor, confusing LCM with HCF), and 540 (three times the LCM, from an arithmetic slip during factorisation).
Checking the answer in five seconds: 180 ÷ 12 = 15, 180 ÷ 15 = 12, 180 ÷ 18 = 10 — all whole numbers, and no smaller multiple passes all three tests, because 90 fails on 12.
You can now decide divisibility, factor and count without dividing. The next lesson puts factorisation to work on the most common quantitative material of all: fractions, decimals and the conversions between fractions, decimals and percentages.