Highest Common Factor

QQ2018789

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Jane has 240 stalks of Roses, 360 stalks of Chrysanthemum and 540 stalks of sunflower.She prepares bouquets of flowers containing the three types of flowers. The number of flowers of each type is the same in all the bouquets.

i) What is the largest possible number of flower bouquets that can be prepared?

ii) How many stalks of Roses,Chrysanthemum,and Sunflower are there in each bouquet?

I got the answer for question i .
The answer is 60 of not mistaken because 60 is the Hcf of 240,360 and 540
However , i am not able to answer Question ii

Your help is much appreciated .Thank You
 
Jane has 240 stalks of Roses, 360 stalks of Chrysanthemum and 540 stalks of sunflower.She prepares bouquets of flowers containing the three types of flowers. The number of flowers of each type is the same in all the bouquets.

i) What is the largest possible number of flower bouquets that can be prepared?

ii) How many stalks of Roses,Chrysanthemum,and Sunflower are there in each bouquet?

I got the answer for question i .
The answer is 60 of not mistaken because 60 is the Hcf of 240,360 and 540
However , i am not able to answer Question ii

Your help is much appreciated .Thank You
Think about it a bit!

You are making 60 boquets

and

there are 360 stalks of Chrysanthemums.

How many Chrysanthemums can you put in each bouquet?
 
Jane has 240 stalks of Roses, 360 stalks of Chrysanthemum and 540 stalks of sunflower.
She prepares bouquets of flowers containing the three types of flowers.
The number of flowers of each type is the same in all the bouquets.

i) What is the largest possible number of flower bouquets that can be prepared?
240 bouquets, each with 1 of each ?

Guess I need another coffee:idea:
 
240 bouquets, each with 1 of each ?
A three-stem "bouquet". It's good you weren't a florist. ;)

I'm pretty sure they want to use as many stems as possible (per bouquet), while maintaining the same stem count for each flower type.

Regarding something else, perhaps you can tell me whether the following question asks for the bouquet stem count or the stem count for each flower type.

ii) How many stalks of Roses,Chrysanthemum,and Sunflower are there in each bouquet?
 
Dunno.
This is specified:
"The number of flowers of each type is the same in all the bouquets"
So why is 240 bouquets not the maximum?
 
… So why is 240 bouquets not the maximum?
Well, for you, it is.

For me, three flowers do not comprise a bouquet. Goggle keywords images bouquet and try to find a picture containing only three flowers. ;)
 
I think we have got off the track of helping the OP.

A prime number is a whole number that cannot be divided evenly by any other whole number except 1 and itself. Every whole number is either a prime or is a multiple of primes.

Let's factor into primes the number of each type of flower:

\(\displaystyle 240 = 24 * 10 = (8 * 3) * (2 * 5) = (2 * 2 * 2) * 3 * (2 * 5) = 2^4 * 3 * 5.\)

Now you can test that 2, 3, and 5 are all primes. So we have found how to express 240 as a product of primes. (Except for order, that expression is unique.)

\(\displaystyle 360 = 36 * 10 = (6 * 6) * (2 * 5) = (2 * 3) * (2 * 3) * (2 * 5) = 2^3 * 3^2 * 5.\)

\(\displaystyle 540 = 54 * 10 = (6 * 9) *(5 * 2) = (2 * 3) * (3 * 3) (2 * 5) = 2^2 * 3^3 * 5.\)

Now any number that is the product of prime factors that are common to a set of numbers will evenly divide each number in that set. So, for example, 5 and 2 are common prime factors to the three numbers, and their product is 10, which does indeed divide evenly into 240, 360, and 540.

Are you with me to here?

Now \(\displaystyle 2^4 = 2 * 2^3 = 2 * 2 * 2^2 = 2 * 2 * 2 \text { and } 3^3 = 3 * 3^2 = 3 * 3 * 3.\)

That's what exponents mean. So \(\displaystyle 2^4\) and \(\displaystyle 2^3\) are not common factors.

but \(\displaystyle 2^2 = 2 * 2\) is. Do you see why?

So what are the common prime factors of 240, 360, and 540?

If you multiply all the common prime factors of a set of numbers, you get their greatest common factor.

So what do you get?
 
I think we have got off the track of helping the OP …

If you multiply all the common prime factors of a set of numbers, you get their greatest common factor.

So what do you get?
I'm not sure why you're asking. In the op, QQ2018789 already stated that they have found the greatest common factor to be 60. QQ2018789 would like help answering question (ii). Subhotosh started that, although I would have referred to the roses.
 
… ii) How many stalks of Roses,Chrysanthemum,and Sunflower are there in each bouquet?

I got the answer for question i … However , i am not able to answer Question ii …
Hello QQ2018789. How do you interpret question (ii)?

I began reading it as, "How many stalks of each flower type are in a single bouquet" because they listed each flower type in the question. But, by the time I reached the question mark, I realized the question is ambiguous; it can also be read as, "How many stalks of flowers are there in each bouquet".

As a student, I dealt with ambiguity in exercises by providing an answer for each possibility:
(ii) If the question asks for the bouquet stalk-count, my answer is ___ .

(ii) If the question asks for the stalk-count for each flower type, my answer is ____ .

:cool:
 
Well, for you, it is.
For me, three flowers do not comprise a bouquet. Goggle keywords images bouquet and try to find a picture containing only three flowers. ;)
Well, true nuff...but this is a math problem: a "minimum" should be specified.
 
I'm not sure why you're asking. In the op, QQ2018789 already stated that they have found the greatest common factor to be 60. QQ2018789 would like help answering question (ii). Subhotosh started that, although I would have referred to the roses.
You may very well be right. I got thrown by the "if not mistaken." I read that as implying that this was a double question, with the first being how to ensure that 60 was indeed the highest common factor.

What now bothers me is where the difficulty comes in answering the second question. It is almost as though the student does not understand why the maximum number of bouquets was dependent on the highest common factor. If that is the source of the difficulty, I am not sure that a Socratic approach will help. But let's give it a shot.

The maximum number of bouquets, each with the same number of each type of flower and with no flower left unused, means that the number of bouquets times the number of flowers of a given type must exactly equal the total number of flowers of that type. It can't be more, or else every bouquet cannot contain the same number of flowers of that type. It can't be fewer, or else some flowers of that type are wasted. But that means that the maximum number of bouquets must be the largest number that divides evenly into the total number of roses and the total number of chrysanthemums and the total number of sunflowers.

Do you understand why?
 
"roses are red
violets are blue"
that's what Jeff said
but it's not true :razz:
 
… It is almost as though the student does not understand why the maximum number of bouquets was dependent on the highest common factor …
AGREE!

… with no flower left unused …
For me, the exercise statement does not infer this. I read the meaning as: make as many bouquets as possible, using as many stalks per bouquet as possible, while maintaining the same stalk count for each flower type.

(I'm wondering how many different answers the instructor will receive.)
 
For me, the exercise statement does not infer this. I read the meaning as: make as many bouquets as possible, using as many stalks per bouquet as possible, while maintaining the same stalk count for each flower type.
You are right again. The problem as worded is underspecified. I added the constraint of all flowers used because that results in a sensible answer. (Perhaps that leads to the same result as your interpretation given that the different types all have a common divisor.) If flowers can be left unused, mere maximization of the number of "bouquets" each containing the same number of each type of flower does indeed suggest 240 "bouquets," an answer of remarkable absurdity for several reasons.

Whether the student has not given the problem completely or the teacher did not specify it fully, perhaps we should wait for clarification from the student.

It is also possible that I am tired today. It has been stressful the last few days.
 
Since post is titled "Highest Common Factor",
perhaps intent is: HCF(240,360,540) = ?

...but a fancy teacher tried to dress it up with flowers :p
 
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