JMC 2020

Its square root

HintAny factor above that point pairs with a smaller one you would already have found.

The why143 needs only 2, 3, 5, 7 and 11 tested (11 × 11 = 121, 13 × 13 = 169), and 11 × 13 = 143 catches it. Odd numbers between 120 and 170 are a classic trap — 143, 161 and 169 all look prime and none is. Try 3 (digit sum), then 7, 11 and 13 before declaring one prime.

Its order of rotational symmetry

HintSpin it and count the matches, including the one at the end.

The whyOrder 2 means a half turn works; order 4, a quarter turn. Every shape has order at least 1. A shape can have rotational symmetry with no line of symmetry, and a rectangle's centre is the point everything turns about.

2

HintOne hop per zero.

The why4.25 × 100 = 425, not 4.2500. Digits move; the decimal point stays put. Converting metres to centimetres is exactly this move, and kilometres to metres is three hops.

1

HintNot prime, not composite — and easy to forget that it counts as a factor.

The why1 is not prime, so it never appears in a prime factorisation, but it is a perfectly good positive whole number. 30 = 2 × 3 × 5 = 1 × 2 × 3 × 5. Watch for the difference between 'prime factors' and 'positive whole numbers' in the wording.

multiply

HintThe word 'of' between two fractions is an operation sign.

The whyTwo thirds of three fifths is 6/15 = 2/5. When a chain of "of"s ends in a known value, reverse it by dividing by each fraction in turn — or, faster, by multiplying by each one turned upside down.

n + 1

HintA 1 by 1 grid already needs a line on both sides.

The whyCells and lines differ by one, like fence panels and fence posts. A 5 × 5 grid has 6 lines each way; each line is as long as the grid, so the wire needed is 2 × (n + 1) × n.

Mean × number of values

HintThe average is fair shares — reverse the sharing.

The whyMean = total ÷ count, so total = mean × count. Most mean questions are really about the total: find it, subtract the parts you know, and what is left belongs to the missing values.

equal

HintThe X shape: look at the angle across the crossing point.

The whyThe two angles on either side of a straight line add to 180°, and the two across a crossing point match. Together with the exterior angle theorem for a triangle, those three facts solve almost every angle chase in the paper.

1/8; 1/5; 3/8

HintWrite each as thousandths or ten-thousandths, then cancel.

The why0.375 is 375/1000, which cancels to 3/8. Moving the decimal point one place divides by 10, so 1.25 = 5/4 and 0.0125 = 1/80 follow at once.

It would be a multiple of 5

HintThink about which times table those last digits always belong to.

The whyA number ending in 0 or 5 divides by 5, and one ending in an even digit divides by 2. So beyond the single digits every prime ends in 1, 3, 7 or 9 — the first filter to apply before testing 3 (digit sum), 7 and 11.

Length × width × height

HintCount the unit cubes in one layer, then the layers.

The whyDoubling one dimension doubles the volume; doubling all three multiplies it by 8. A cuboid that is 2, 2 and 5 times the size of a cube in its three directions holds 2 × 2 × 5 = 20 times as many unit cubes.

210

HintSmall primes, each used once.

The whyAny number that is a multiple of 2, 3, 5 and 7 is a multiple of 210. Adding 1 to a product of primes is the heart of Euclid's proof that primes never run out: the result cannot be divided by any prime you multiplied, so some prime is still missing from your list.