JMC 2025

Its primes fit inside

HintThink building blocks, not long division.

The whyEvery prime in the smaller number must appear in the larger at least as many times. 27 = 3³ is a factor of 3⁴ × 5² because three 3s are available; 50 = 2 × 5² is not, because there is no 2. One factorisation checks a whole list of candidates at once.

Three quarters

HintThe named fraction and the fixed part together make the whole.

The why"Plus a quarter of itself" leaves the other three quarters as the fixed part, so the whole rope is 21 ÷ 3 × 4 = 28 cm. Whenever a quantity is described as "a fixed amount plus a fraction of itself", the fixed amount is the remaining fraction of the whole.

Round and estimate

HintThe options are spaced widely enough that exact working is wasted time.

The why97 × 31.8 is close to 100 × 30 = 3000, which is all you need to pick between choices that differ by a thousand. Look at the spread of the options before you decide how precisely to calculate — the spread is a hint the setter has given you.

A square number

HintFactors usually come in pairs, unless one pair is a repeat of itself.

The whyFactors pair up (2 with 18 for 36) except when a factor pairs with itself (6 × 6), so only squares have an odd count. 36 has nine factors: 1, 2, 3, 4, 6, 9, 12, 18, 36. A question saying "an odd number of factors" is quietly saying "a square".

Smaller than N

HintIf a whole extra N were still left over, the division would not be finished.

The whyA remainder of 3 on dividing by N means N is bigger than 3 — and N is a factor of the number minus 3. So "which N leave remainder 3?" means: list the factors of (number − 3) and keep only those above 3.

0

HintWhich digit would make a number shorter than it looks?

The whyA leading nought is simply not written. So when you build the smallest number from given digits, the smallest NON-ZERO digit goes first and 0 comes second — 1023 beats 0123, which is really the three-digit 123.

20

HintPer cent means out of a hundred.

The whyOnce 10% is known, 5% is half of it, 20% is double, 30% is triple. Nested percentages multiply: 30% of 50% of an amount is 0.3 × 0.5 = 0.15 of it, so chains of "of" reduce to one multiplier.

The digit sum is a multiple of 3

HintYou never need to divide — the answer hides in the digits.

The why741 has digit sum 12, so it divides by 3; 742 does not. The same test with 9 finds multiples of 9. Contrast the test for 4, which looks only at the last two digits — 916 divides by 4 because 16 does.

a² + b² = c²

HintNamed after a Greek; a right angle is required.

The whyA tilted square drawn inside a bigger square has its side as the hypotenuse of a corner triangle, so its area is the sum of two squares — no square root needed. With legs 2 and 3 the tilted square has area 13, whatever unit you use.

45

HintIts square ends in 25, so it ends in 5 — and it is under 50.

The whySetters love the year: 2025 = 45² = 3⁴ × 5², so its factors include 9, 27, 81 and 405. Before any Challenge, know whether that year’s number is prime, a square, or a multiple of small primes — the first question often plays with it.

12

HintThink about one face first, then count the faces.

The whyAll twelve are the same length — each is the hypotenuse of a right-angled triangle whose legs are two edges. Space diagonals, from a corner straight through the middle to the opposite corner, are different: there are only four, and they are longer.

8; 27; 64; 125

HintNot the squares — one power higher.

The whyKnow these as well as the squares. Among the one-digit numbers only 1 and 8 are cubes; 27 and 64 are the two-digit ones; and 27 + 64 + 125 = 216 = 6³ is a famous coincidence worth recognising on sight.