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We had lots of correct solutions to this
problem and many of you explained your thinking very clearly. Well
Musab from Orchard School answered the first
part of the problem:
Ben might have started with a number 1 more than a multiple of 3
because it says when Ben put the garlic into groups of 3 there was
1 garlic plant left. For example Ben could have started with 7.
You're right, Musab. What other possibilities
could there have been then? Pupils at St Mary's in Tetbury
approached the second part of the problem in a very logical (or
systematic) way. They said:
We decided to work out all the numbers it could be.
We started with 6 rows with one left over, and wrote down all the
multiples of 6, add 1, up to 100.
Then we looked at 5 rows with one left over, and wrote down all the
multiples of 5, add 1, up to 100.
We rubbed out all the numbers that weren't in both.
Then we looked at 4 rows with one left over, and wrote down any
multiples of 4, add 1, that appeared in our lists for 5 rows and 6
Then we looked at 3 rows with one left over, and there was only one
number that worked for all of them. We think the solution is
PJ, Josh and Michael tried a few things out
and gradually got close to the answer. This can be a very useful
way of solving a problem (we call it trial and improvement):
We started by just trying numbers. Then Josh started doing
multiples, and we found out some numbers that have one (in the units column?) work so we tried 41 then
51 then we found that 61 worked, and that's how we got our answer:
Qiuying of Wimbledon High School explained the
solution in slightly higher level maths. (Don't worry if you
haven't come across these ideas yet.)
Less than 100:
The Number of garlic cloves equals the Lowest Common Multiple
of 2, 3, 4, 5, 6 and then plus 1.
LCM = 60 So the Number is 60 + 1 = 61
The first one more than 100:
The numbers of garlic cloves equals 60 x 2 + 1 = 121