Consecutive Numbers

An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.

Calendar Capers

Choose any three by three square of dates on a calendar page...

Days and Dates

Investigate how you can work out what day of the week your birthday will be on next year, and the year after...

Tricky Customer

Age 11 to 14 ShortChallenge Level

Listing the mutliples
Multiples of 3:            (99),      102,        105,        ... ,        150
Position relative to 99:(99), 99 + 1$\times$3, 99 + 2$\times$3, ..., 99 + 51 = 99 + 17$\times$3
Count:                                       1,              2,      ... ,                        17

Multiples of 5:               100,       105,           110,       ... ,       150
Position relative to 100: 100, 100 + 1$\times$5, 100 + 2$\times$5, ... , 100 + 10$\times$5
Count:                                           1,               2,      ... ,           10
11

17 + 11 = 28 numbers not allowed
But some numbers (like 105 and 150) are in both lists! They are multiples of 3 and 5 ie multiples of 15

105, 120, 135, 150 are counted twice

28$-$4 = 24 numbers not allowed
Out of 51 houses altogether
51$-$24 = 27 houses allowed.

Counting the multiples
There are $51$ houses numbered from $100$ to $150$ inclusive. Of these, $17$ are multiples of $3$, eleven are multiples of $5$ and four are multiples of both $3$ and $5$. So the number of houses Charlie can choose from is $$51-(17+11-4) = 27.$$

This problem is taken from the UKMT Mathematical Challenges.
You can find more short problems, arranged by curriculum topic, in our short problems collection.