GCF and LCM Word Problems | PDF | Leisure | Nature - Free Printable

1. The maximum number of bracelets Sally can make is 5. This is found by calculating the GCF of 10, 25, and 30, which is 5.

2. The maximum number of packages the Student Council can make is 15. This is found by calculating the GCF of 60, 90, and 45, which is 15.

3. The maximum number of identical baskets the PTA can make is 25. This is found by calculating the GCF of 100, 50, and 75, which is 25.

4. Over the next 40 days, Robbie and Cynthia will visit their grandmother on the same day 2 times. This is found by calculating the LCM of 6 and 5, which is 30. Within 40 days, the common visit days are day 30 and day 60 (which is beyond 40), so only day 30 counts. Wait, let's recalculate: LCM(6,5)=30. In 40 days, they meet on day 30 only once? But 30 is within 40, and 60 is not. So it should be 1 time? Let me check again. Actually, LCM(6,5)=30, so they meet every 30 days. Starting from day 0 (if they both visited today), they meet again on day 30. Since the question says "over the next 40 days", and assuming they start counting from day 1, then day 30 is within 40. Is there another day? 30*2=60>40, so only one day. But the answer might be 1? However, sometimes problems consider the starting point. Let me re-read: "Over the next 40 days". If they both visited today (day 0), then next common day is day 30. So in the next 40 days (days 1 to 40), day 30 is included. Only one day. But perhaps the problem means including day 0? The problem doesn't specify if they visited today. Typically, "over the next 40 days" means from now to 40 days later. If they visited today, then the next common visit is day 30, and that's one time. But let's see the standard interpretation. Actually, LCM(6,5)=30, so the common visits occur every 30 days. In a 40-day period, there is only one such day (day 30). So the answer should be 1. But I think I made a mistake earlier. Let me correct: the answer is 1.

Wait, no: if they both visit on day 0, then they visit together on day 0, 30, 60, etc. "Over the next 40 days" might include day 0 or not? Usually, "over the next 40 days" means from day 1 to day 40, excluding day 0. So only day 30. So 1 time. But let me confirm with the problem. It says "over the next 40 days", and they visit every 6 and 5 days. The first common day after day 0 is day 30. So yes, 1 time.

But I recall that sometimes these problems count the starting point. To be safe, let's calculate: the common multiples of 6 and 5 up to 40 are 30. 0 is also a common multiple, but if "next 40 days" starts from tomorrow, then day 0 is not included. So only day 30. Answer is 1.

Actually, upon second thought, the problem likely assumes they start visiting from day 1, and we need to find how many days within 1 to 40 are common multiples. LCM is 30, so day 30 is the only one. So 1.

But let me see if there's a trick. Perhaps they visited on different days initially? The problem doesn't say when they last visited. It says "Robbie visits every 6 days, Cynthia every 5 days." It doesn't specify if they started on the same day. But typically in such problems, we assume they start at the same time, i.e., day 0. And "over the next 40 days" means from day 1 to day 40. So only day 30. Answer is 1.

I think I was overcomplicating. The standard answer for such problems is to find LCM and see how many multiples fit in the range. LCM=30, multiples: 30, 60, ... Only 30 is <=40, so 1 time.

5. The next day Peggy will water all three plants on the same day is October 31st. This is found by calculating the LCM of 5, 3, and 2, which is 30. Since she watered them all on October 1st, adding 30 days brings us to October 31st.

6. Of the 20 bags, 1 bag has all three items. This is found by calculating the LCM of 3, 4, and 5, which is 60. Since 60 > 20, there is no bag within the first 20 that is a multiple of 60. But wait, that can't be right because the LCM is 60, meaning every 60th bag would have all three, but since we only have 20 bags, none should have all three? But that seems odd. Let me think.

The items are placed in every 3rd, 4th, and 5th bag. So a bag has all three items if its number is divisible by 3, 4, and 5, i.e., by LCM(3,4,5)=60. The multiples of 60 within 1 to 20 are none, since 60>20. So the answer should be 0.

But that seems counterintuitive. Perhaps I misinterpreted. Let me read: "in every 3rd bag he put lollipops" — so bags 3,6,9,12,15,18. "in every 4th bag" — bags 4,8,12,16,20. "in every 5th bag" — bags 5,10,15,20. Now, which bags have all three? We need bags that are multiples of 3,4, and 5. So common multiples of 3,4,5 up to 20. LCM is 60, as said, and no multiple of 60 is <=20. But let's list the bags that have lollipops: 3,6,9,12,15,18. Hershey Kisses: 4,8,12,16,20. Fake eyeballs: 5,10,15,20. Now, intersection: look for bags in all three lists. Bag 12: in lollipops (yes, 12÷3=4), in Hershey (12÷4=3), in eyeballs? 12÷5=2.4, not integer, so no. Bag 15: lollipops (15÷3=5), eyeballs (15÷5=3), but Hershey? 15÷4=3.75, not integer. Bag 20: lollipops? 20÷3≈6.67, not integer. Hershey: 20÷4=5, yes. Eyeballs: 20÷5=4, yes. But not lollipops. Bag 6: lollipops yes, Hershey? 6÷4=1.5, no. Bag 10: eyeballs yes, lollipops? 10÷3≈3.33, no. Bag 18: lollipops yes, others? 18÷4=4.5, no; 18÷5=3.6, no. Bag 4: Hershey yes, others? 4÷3≈1.33, no; 4÷5=0.8, no. So no bag has all three items. So the answer is 0.

But that seems surprising. Perhaps I need to check if there's any bag that is multiple of lcm. LCM(3,4,5)=60, and 60>20, so no. Yes. So for problem 6, the answer is 0.

Now, to summarize:

1. 5
2. 15
3. 25
4. 1
5. October 31st
6. 0

But for problem 4, I think I have a mistake. Let me double-check. Robbie visits every 6 days, Cynthia every 5 days. If they both visited on day 0, then they visit together on days that are multiples of LCM(6,5)=30. So days 0, 30, 60, etc. "Over the next 40 days" — if this means from day 1 to day 40, then only day 30 is included. So 1 time. But sometimes "over the next 40 days" might include day 0, but that doesn't make sense because "next" usually excludes the present. To be precise, let's assume the visits start from day 1. The first common visit is at day 30. So in 40 days, only one common visit.

Perhaps the problem means how many times they visit on the same day within the next 40 days, including if they visit on day 0. But the problem doesn't say they visited today. It just says they visit every 6 and 5 days. Without a starting point, we assume they start at the same time, say day 0. Then the common visits are at days 0, 30, 60, etc. If "over the next 40 days" means from day 0 to day 40, then days 0 and 30 are included, so 2 times. Ah, that might be it. In many such problems, "over the next X days" includes the starting day if it's a common visit.

Let me see the wording: "Over the next 40 days, how many will the two cousins visit their grandmother on the same day?" It doesn't specify if they visited today or not. But typically, in GCF/LCM word problems, if not specified, we assume they start together at time zero, and "over the next X days" means from day 0 to day X inclusive or exclusive? This is ambiguous.

To resolve, let's think: if they both visit on day 0, and we consider the next 40 days as days 1 through 40, then only day 30. But if we consider the period from day 0 to day 39, then day 0 and day 30 are both within 40 days (since 0 to 39 is 40 days). Day 0 to day 39 inclusive is 40 days. So days 0 and 30 are both in this range. So 2 times.

I think this is the intended interpretation. Because if we take days 1 to 40, it's 40 days, but day 0 is not included. But "over the next 40 days" often includes the current day if it's a visit day. In many textbooks, for such problems, they include the starting point.

For example, if two events happen every 2 and 3 days, and they start together, over the next 6 days, they coincide on day 0 and day 6, so 2 times.

So for this problem, LCM=30, common days: 0, 30, 60, ... Within 0 to 39 (40 days), days 0 and 30 are included. So 2 times.

Yes, I think that's correct. So answer for 4 is 2.

Similarly, for problem 6, I think 0 is correct.

So final answers:

1. 5
2. 15
3. 25
4. 2
5. October 31st
6. 0