Factors of 12: 1, 2, 3, 4, 6, 12 / of 18: 1, 2, 3, 6, 9, 18. In both: 1, 2, 3, 6 → GCD 6

GCD 4; LCM: 8, 16, 24 / 12, 24 → 24

Factors of 15: 1,3,5,15 / of 20: 1,2,4,5,10,20 → 5

LCM 18. 54 is a common multiple but not the lowest. 9 is not a multiple of 6, so counting up the multiples of 6 you meet 18 first.

GCD of 18 and 24 is 6 → side 6 cm. (18÷6) x (24÷6) ＝ 3 x 4 ＝ 12 squares

LCM of 12 and 18 → 36 seconds

LCM of 4 and 6 is 12 → 12, 24, 36, 48

Product ＝ 8 x 48 ＝ 384. Write them as 8a and 8b with a, b coprime; 64ab ＝ 384 → ab ＝ 6 → (1,6) or (2,3). 8 and 48, or 16 and 24

GCD 8, LCM 48

LCM of 5 and 6 is 30 → 30, 60

GCD of 36 and 24 is 12 → 12 people; 36÷12 = 3 pencils and 24÷12 = 2 erasers each

Question 5 is the GCD, question 6 the LCM.

Splitting things up is the GCD; waiting for things to line up is the LCM. That sentence is what the week was for.

LCM of 3, 4, 6 is 12

1920÷120＝16, 1080÷120＝9 → 16 : 9

1024÷256＝4, 768÷256＝3 → 4 : 3

1170÷6＝195, 2532÷6＝422 → 195 : 422 (taller than wide)

16÷9＝1.77…, 4÷3＝1.33… → 16 : 9 is wider. The bigger width÷height is, the wider the shape.

GCD of 3840 and 2160 is 240 → 16 : 9. Same shape

GCD of 6000 and 4000 is 2000 → 3 : 2

1280÷80 = 16, 720÷80 = 9 → 16 : 9

2048÷512 = 4, 1536÷512 = 3 → 4 : 3

1440 ÷ 9 = 160, then 16 x 160 = 2560

16 : 9 is wider. Put both over a height of 9: 4 : 3 becomes 12 : 9, and 16 > 12.

They are the same thing — both divide through by the greatest common divisor.

1920 : 1080 → 16 : 9 and ¹⁰⁸⁰⁄₁₉₂₀ → ⁹⁄₁₆ are one calculation. Seeing that Friday is only a change of notation makes the whole unit much lighter.

2560÷16＝160, 9x160＝1440

³⁄₆ ＋ ²⁄₆ ＝ ⁵⁄₆

⁹⁄₁₂ ＋ ²⁄₁₂ ＝ ¹¹⁄₁₂

¹⁵⁄₁₈ − ⁴⁄₁₈ ＝ ¹¹⁄₁₈

GCD 12 → ²⁄₃

³⁄₁₂ ＋ ¹⁄₁₂ ＝ ⁴⁄₁₂ → ¹⁄₃

¹⁶⁄₂₄ vs ¹⁵⁄₂₄ → ²⁄₃ is larger

⁹⁄₁₂ − ²⁄₁₂ ＝ ⁷⁄₁₂ m

Accept anything with this sense: '¹⁄₂ alone is bigger than ²⁄₅, so the total cannot shrink. The denominator says how big one piece is, so adding denominators changes the pieces.'

Both work. 48 makes the numbers larger and leaves more cancelling at the end, so the lowest common multiple is the usual choice.

Over 24 → ¹⁵⁄₂₄ + ¹⁴⁄₂₄ = ²⁹⁄₂₄ = 1⁵⁄₂₄

Over 36 → ²⁷⁄₃₆, ³⁰⁄₃₆, ²⁸⁄₃₆ → ⁵⁄₆ > ⁷⁄₉ > ³⁄₄

Over 12 → ⁸⁄₁₂ − ³⁄₁₂ − ²⁄₁₂ = ³⁄₁₂ = ¹⁄₄ L

³⁄₆＋²⁄₆＋¹⁄₆ ＝ ⁶⁄₆ ＝ 1. Noticing it comes to exactly 1 is enough.

LCM of 12 and 18 → 36 teeth

36÷12＝3 turns, 36÷18＝2 turns. The ratio of turns, 3 : 2, is the reverse of the ratio of teeth, 12 : 18.

Sense required: '216 is a common multiple but not the lowest. 18 is not a multiple of 12, so counting multiples of 12 reaches 36 first.'

LCM of 8 and 12 → 24 teeth

48 ÷ 16 ＝ 3 turns

50 ÷ 3 ＝ 16 remainder 2. Remainder 1 red, 2 blue, 0 yellow → blue

100 ÷ 3 = 33 remainder 1 → the first colour → red

60 ÷ 4 = 15 remainder 0 → green, the fourth

A remainder of 0 means a full cycle has just finished, so it is the last colour — green here.

Answering "the first" for a remainder of 0 is the commonest mistake in these questions.

29 − 1 = 28 days later; 28 ÷ 7 = 4 remainder 0 → Thursday

The remainder of a division.

Gears ask when things line up, which is the LCM; cycles ask which one you land on, which is the remainder. Both are about repetition, but they ask different questions.

LCM of 12, 18, 36 → 36 teeth