A school has lockers numbered 1 to 200. The caretaker sticks coloured dots on the lockers using these rules: • A locker number that is a multiple of 3 and a multiple of 5 gets a gold dot only. • A locker number that is a multiple of 3 but not a multiple of 5 gets a red dot. • A locker number that is a multiple of 5 but not a multiple of 3 gets a blue dot.
- How many lockers numbered from 100 to 200, including 100 and 200, get a gold dot?
- Among the lockers numbered 1 to 150, how many get either a red dot or a blue dot?
Need a hint?
Lockers that are multiples of both 3 and 5 are gold, so take them out of the multiples of 3 and of 5 before you add.
Worked answer
- (a) A gold dot needs a multiple of both 3 and 5, so a multiple of 15.
- The first multiple of 15 that is at least 100 is 105. The last one not more than 200 is 195.
- 105, 120, 135, 150, 165, 180, 195 gives 7 lockers.
- (b) Multiples of 3 from 1 to 150: 150 ÷ 3 = 50. Multiples of 5: 150 ÷ 5 = 30. Multiples of 15: 150 ÷ 15 = 10.
- Red dots: 50 − 10 = 40. Blue dots: 30 − 10 = 20.
- Red or blue: 40 + 20 = 60 lockers.
The mistake this question tests
Factors and multiples (HCF, LCM)
Many pupils just add the multiples of 3 and the multiples of 5, which gives 80, so the gold lockers are counted in too. The gold lockers get neither a red nor a blue dot.
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