Growing garlic
Ben and his mum are planting garlic. Can you find out how many cloves of garlic they might have had?
Problem
Ben is on the allotment with his mum. They would like to grow some garlic and are deciding how to plant the garlic cloves.
Ben arranges the cloves into three equal rows and finds that he has one spare clove.
How many cloves might he have had to start with?
How do you know?
Ben plants cloves of garlic in two equal rows and has one clove left over. So he tries again.
He plants cloves in three equal rows and has one left over. So he tries again.
He plants cloves in four equal rows and has one left over. So he tries again.
He plants cloves in five equal rows and has one left over. So he tries again.
He plants cloves in six equal rows and still has one left over.
We know that he has fewer than 100 garlic cloves. How many did he have?
How do you know?
How many cloves might he have had if there were more than 100?
You might like to use this interactivity to try out your ideas:
Getting Started
Have you tried out your ideas using the interactivity? Or you could use some counters or cubes to stand for the garlic cloves and arrange them in rows.
If Ben had only planted one clove in each row, how many cloves would he have had?
What other numbers of cloves could he have planted in each row? So how many cloves would he have had altogether then?
Student Solutions
Well done to everybody who worked out that Ben planted 61 cloves of garlic. The solutions below explain different ways of working this out.
Dana from Twyford School in the UK went down from 100, checking the different possibilities for 3, 4, 5 and 6 rows until she found a number that worked in each case. She said:
I found the highest multiple of three that was under 100 which was 99 and then I added one onto that as there is always 1 left. This made 100. Then I did this for the rest of the multiples of 3. I did this for 4 rows which was 96 so I added one onto that and I had 97. Then I did it for 5 and 6 rows and when I had done this I noticed that 61 garlic cloves was on the list for 3,4,5 and 6. This is how I found out that Ben has 61 garlic cloves.
With 3 rows you can have:
100 garlics
97 garlics
94 garlics
91 garlics
88 garlics
85 garlics
82 garlics
79 garlics
76 garlics
73 garlics
70 garlics
67 garlics
64 garlics
61 garlics
With 4 rows he can have:
97 garlics
93 garlics
89 garlics
85 garlics
81 garlics
77 garlics
73 garlics
69 garlics
65 garlics
61 garlics
With 5 rows he can have:
96 garlics
91 garlics
86 garlics
81 garlics
76 garlics
71 garlics
66 garlics
61 garlics
With 6 rows he can have:
97 garlics
91 garlics
85 garlics
79 garlics
73 garlics
67 garlics
61 garlics
Ben has 61 garlic cloves altogether.
Well done Dana!
Dev from Dragon School in the UK also looked for a number that was one more than a multiple of 2, 3, 4, 5 and 6. They said:
The points I know
1) The number is less than 100
2) The total number of cloves is going to be odd because when you plant it in rows of 2, one is left over, which tells us the total number of cloves = even number + 1
3) This means total number of cloves = (multiple of 2,3,4,5,6) + 1
4) The number which is a multiple of 2,3,4,5 and 6 has to end in a 0 because it is an even number and multiple of 5
5) The even number which fits all the above conditions is 60
6) 60 is a multiple of 2,3,4,5 and 6
7) Total number of cloves Ben planted was 60+1=61
Well done for narrowing down the different possibilities, Dev. This question becomes easier when you think about finding a number which is a multiple of 2, 3, 4, 5 and 6, and then adding one to that number, so well done for spotting that method!
Ethan from Churchfields also thought about the possible ones digit of the number. He said:
If a number is one more than a multiple of 5 it has to end in a 1 or a 6. If the number ended in 6 it would be even so there would be none left over when they were planted in rows of 2.
I took all the two digit numbers below 100 that end in 1 (1 is too small) and crossed out the ones that were multiples of 3, or would have a remainder of 2 when divided by 3.
This left me with 31, 61 and 91. I then worked out what would happen if I divided them by 4. The only one with a remainder of 1 was 61 so that must be the answer.
Good work, Ethan - this is a really quick method as there aren't many numbers below 100 that end in 1.
Dhruv from St Anne’s RC Primary School in the UK used a combination of Dev and Ethan's ideas. He sent in this explanation:
Well done Dhruv!
Arjun from Nascot Wood Junior School in the UK also used the fact that 60 is a common multiple of 2, 3, 4, 5 and 6. He said:
Since there must be 1 garlic clove left after the cloves have been sorted into 2, 3, 4, 5 and 6 rows, we must find the Lowest Common Multiple (LCM) of 2, 3, 4, 5 and 6, then add 1. You will find that the LCM is 60. 60 + 1 = 61. So the lowest amount of cloves that he can have is 61.
For above 100, the lowest multiple of 60 that is above 100 is 2 x 60 = 120. But we have to add 1, so the lowest amount of cloves that is above 100 is 121.
Good work, Arjun. I wonder what other possible amounts of cloves Ben might have had if there were more than 100?
We were also sent some good solutions by Ivan and Kenny at Sahuaro Elementary in the USA, and by Luca and Toby at Pierrepont Gamston Primary School in England. Thank you all for sending us your thoughts about this task.
Teachers' Resources
Growing Garlic
Ben is on the allotment with his Mum. They would like to grow some garlic and are deciding how to plant the garlic cloves.
Ben arranges the cloves into three equal rows and finds that he has one spare clove.
How many cloves might he have had to start with?
How do you know?
Ben plants cloves of garlic in two equal rows and has one clove left over. So he tries again.
He plants cloves in three equal rows and has one left over. So he tries again.
He plants cloves in four equal rows and has one left over. So he tries again.
He plants cloves in five equal rows and has one left over. So he tries again.
He plants cloves in six equal rows and still has one left over.
We know that he has fewer than 100 garlic cloves. How many did he have?
How do you know?
How many cloves might he have had if there were more than 100?
You might like to use this interactivity to try out your ideas:
Why do this problem?
The idea of this problem is to encourage children to identify and describe a pattern, and to extend this into a general rule. Depending on their experience, they can also be encouraged to use relevant vocabulary associated with factors and multiples.
Possible approach
This problem featured in the NRICH Primary webinar in April 2022.
You could use this problem with the whole class as it is an example of a low threshold high ceiling activity - accessible to all but also having in-built challenge for the most confident.
You may want to start by asking some simpler questions, for example:
Ben has 15 bulbs of garlic.
- Can he plant them in two equal rows? Why or why not?
- Can he plant them in three equal rows? Why or why not?
- Can he plant them in four equal rows? Why or why not?
- What else could you tell me?
In each case, encourage learners to use practical resources, to show their solution.
Then, ask the children to find how many bulbs Ben could have if he plants them in three equal rows but has one left over, as in the first part of the problem. Have cubes, counters, bottle tops, squared paper, hundred squares... available for everyone to use, should they wish. List children's suggestions and use the interactivity, or counters/magnets on the board, to check them. The interactivity offers a visual representation of factors, multiples and remainders by arranging numbered garlic cloves into a rectangular array, i.e. into rows of equal size). The 'left over' cloves are circled in red.
Rearrange the solutions in order of size on the board and ask learners what they notice. Encourage comments which indicate the children have identified the pattern ("It's one more than the three times table") rather than them offering specific examples that 'fit' (it could be 22, 4, 31...). Many will make comments about the "numbers going up in threes", which is absolutely correct and an excellent starting point. However, 'one more than the three times table', or 'one more than a multiple of three' allows us to identify instantly whether a particular number is in the sequence, as opposed to having to list all the numbers in order, going up in threes.
Pose the rest of the problem and leave the class to pursue their ideas, working in pairs. You may wish them to have access to the interactivity in pairs as well as access to a range of practical resources. Emphasise that you are interested in their answer or answers, but you are particularly interested in how they go about finding solutions. As they work, look out for pairs who have developed useful ways of recording their thinking. Some learners will be recording to help them find a solution (e.g. drawing blobs for the garlic bulbs), some will be recording to keep track of what they have tried - what has worked and what has not! Use a mini plenary to draw attention to some of the different ways of recording and encourage children to share their reasons for recording in this way.
Bring everyone together to share solutions and help them to differentiate between specific answers and general ones ("If there is one left over with two rows it must be an odd number", rather than "It must be 21 or 15 or 7"). It is interesting to note that there are far fewer multiples of 6 between 1 and 100 than multiples of 2, so starting with numbers that are one more than a multiple of 6 would give fewer possibilities compared with starting with odd numbers. Look out for learners that realise this, thus demonstrating their deep understanding of the number system.
Key questions
Possible support
Physical apparatus helps to consolidate the idea of 'one more than a multiple'. Listing possible answers in order of size can also help the children to spot patterns and encourages them to work systematically.
Possible extension
Children could create their own question with a unique answer for a partner to solve.