Factors and multiples in word problems

A baker has fewer than 70 buns. She can pack all of them into boxes of 6 with none left over, or into boxes of 8 with none left over. What is the greatest number of buns she can have? Try it first. Pause if you like. We do not know the total number of buns. It must pack exactly in sixes and in eights, so it must be on both lists of multiples. List the multiples of 6 that are less than 70. Now list the multiples of 8 that are less than 70. Which numbers are on both lists? We need the greatest one below 70. 24 and 48 are the common multiples of 6 and 8 that are less than 70. 48 is twice 24. Here are two groups of 24 buns. These 48 buns fill boxes of 6 with none left over. Keep every bun. Repack the same 48 in boxes of 8. Again, none are left over. The greatest number of buns she can have is 48. Both packing rules work, and 48 is the greatest shared number below 70.

Boon Keat bought an equal number of pens and rulers. Pens are sold in packs of 6 and rulers are sold in packs of 9. What is the smallest possible number of packs of pens he bought? Try it first. Pause if you like. The number of pens and the number of rulers must be equal. Count on in each pack size until the two lists first meet. Pens come in packs of 6. Their possible totals start 6, 12, 18, and so on. Rulers come in packs of 9. Their possible totals start 9, 18, and so on. Which is the smallest shared number? Then remember: the question asks for packs of pens. 18 is 3 times 6 and also 2 times 9. We can buy 18 pens and 18 rulers. Arrange those pens in packs of 6, and those rulers in packs of 9. The two totals stay equal. 18 divided by 6 is 3. So there are 3 packs of pens. The smallest possible number of packs of pens he bought was 3. Read what the question asks: the number of items, or the number of packs.