What is the greatest number of identical bouquets that can be made out

SOLUTION

What is the greatest number of identical bouquets that can be made out of 21 white and 91 red tulips if no flowers are to be left out? (Two bouquets are identical whenever the number of red tulips in the two bouquets is equal and the number of white tulips in the two bouquets is equal)

(A) 3
(B) 4
(C) 5
(D) 6
(E) 7

Since no flowers are to be left out, then the number of bouquets must be a factor of both 21 and 91. For example, we cannot have 2 bouquets since we cannot divide 91 red tulips into 2 bouquets without one tulip left over.

Only answer choice which is a factor of 91 is E (7).

Answer: E.
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Inorder to find the greatest no of identical bouquets we have to find out the Greatest common divisor of 21 and 91

Prime Factors of 21 - 7x3
Prime Factors of 91 - 7x13

The greatest common factor is 7. Hence the maximum no of identical bouquets that can be made is 7. Answer - Option (E)

Only 7 is divisible with 21 and 91, So, that should be the answer..

Is this the right way to solve?

7 is NOT divisible by either 21 or by 91. 7 is a factor of both 21 and 91.

P.S. Solutions will be published on weekend.
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Oops.!.. Ya My bad...

Thanks Bunuel..

Will wait for solutions...

Option E.
21=7*3
91=7*13
If we have 7 bouquets,each will have 3 white tulips and 13 red tulips.So they will be identical.

Hi All,

This question is essentially about prime-factorization, but you can actually avoid some of the work if you're comfortable with basic division.

We're told to make IDENTICAL bouquets using 21 white and 91 red tulips AND we're told that NO flowers are to be left out.

With 21 white tulips, there are only three ways to form identical bouquets:

21 bouquets with 1 white flower each
7 bouquets with 3 white flowers each
3 bouquets with 7 white flowers each

With 91 red tulips, we just have to see which of those options divides into 91...

21 does NOT divide evenly into 91
7 DOES divide evenly (13 times)
3 does NOT divide evenly into 91

We're asked to find the GREATEST number of possible bouquets that can be formed, but since we're making multiple bouquets, there's only one answer that fits.

Final Answer:

GMAT assassins aren't born, they're made,
Rich

But is it a right approach to calculate GCF of 21 and 91 (which is 7) and therefore in my opinion also the greatest number of identical buquets? Or just by coincidence the same result? Answers appreciated!

Hi noTh1ng,

Calculating the GCF is one approach to solving this question, although there are others. My approach was based on listing out the possibilities and finding the one that matched all of the information in the prompt; you could also have used the answer choices to your advantage and eliminated the numbers that did not match the info in the prompt.

GMAT questions are always carefully designed (even the wrong answers are carefully chosen), so there are almost never any coincidences on Test Day. Sometimes a subtle Number Property or other pattern occurs in a question, and you don't necessarily need to know the rule or pattern to get the correct answer, but the existence of the pattern is never by accident.

GMAT assassins aren't born, they're made,
Rich

Dear noTh1ng

It is not a coincidence.

To understand this, let's assume that we make n identical bouquets, in each of which:

Number of white flowers = w
Number of red flowers = r

Since we are given that no flower is to be left out, we can write:
wn = 21
And, rn = 91

That is, w = 21/n
And, r = 91/n

Now, w and r, being the number of flowers, have to be integers (because we cannot have 2.5 or 3.3 flowers in a bouquet)

So, n has to be a number that completely divides 21 as well as 91.

But the question asks us to find the greatest possible value of n.

That is, the greatest possible number that completely divides 21 as well as 91.

What is this number known as? Greatest Common Factor

Hope this helped clarify your doubt.

Best Regards

Japinder
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"What is the greatest number of identical bouquets that can be made out of 21 white and 91 red tulips if no flowers are to be left out? (Two bouquets are identical whenever the number of red tulips in the two bouquets is equal and the number of white tulips in the two bouquets is equal.)"

What's to stop you from creating 21 bouquets, each containing 1 white tulip and 1 red tulip, and 1 bouquet containing 70 red tulips? In that scenario, you have 21 identical bouquets.

Hi mh72,

In your 'scenario', you've left out 70 red tulips from the 21 bouquets that you made - but the prompt says that NO flowers are to be left out. Thus, your solution does not match the specific restrictions that the prompt gave you to work with.

GMAT assassins aren't born, they're made,
Rich

Hi All,

This question is essentially about prime-factorization, but you can actually avoid some of the work if you're comfortable with basic division.

We're told to make IDENTICAL bouquets using 21 white and 91 red tulips AND we're told that NO flowers are to be left out.

With 21 white tulips, there are only three ways to form identical bouquets:

21 bouquets with 1 white flower each
7 bouquets with 3 white flowers each
3 bouquets with 7 white flowers each

With 91 red tulips, we just have to see which of those options divides into 91...

21 does NOT divide evenly into 91
7 DOES divide evenly (13 times)
3 does NOT divide evenly into 91

We're asked to find the GREATEST number of possible bouquets that can be formed, but since we're making multiple bouquets, there's only one answer that fits.

Final Answer:

GMAT assassins aren't born, they're made,
Rich

This is a case of a question u don't understand but ended up getting it right.
only 7 is a factor of 21 and 91

Hey chetan2u
I am having trouble understanding this Question.
I have seen all the solutions above,but none of them is making sense to me.
You mind solving this one.


Here is what i did
Given => 21 white and 92 red tulips

So to make identical bouquets we can make the arrangement as follows =>
1 each for 21 bouquets in which
2 each for 10 bouquets and and one in the last one.
And we can continue this way to make infinite arrangements.
I understand the use the word "identical" but if w give 2 each => we can make 10 identical bouquets and one bouquets with leftover flowers right ?
What i am missing here?
It feels like a pretty difficult Question to me.

Regards
Stone Cold
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Greatest no of Identical Bouqets of 21 white tulips and 91 red tulips is HCF of 21& 91 = 7

Now, Check...

Each Boquet must have 3 ( ie, 21/7) White tulips and 13 ( ie, 91/7) red tulips , so all tulips are utilized..

Hence, correct answer will be (E)

stonecold hope this helps...
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Hi,
What the Q means is that each bouquet is identical so each should have same number of white,w , and same number of red, say r..
Now w has to be a factor of 21 , since if you take 2 the last bouquet will have only 1, thus all will not be identical.
Only w as 1, 3, 7 or 21 will fit in..

Similarly r should be a factor of 91, so 1, 7,13 or 91..
Example each has7,7..., so 91/7=13 bouquets of 7 red, OR each has 13, so 7 bouquets of 7 each...

Now the MAIN point comes out is both white and red have to be divided in same number of bouquets ...
So only identical are 1and 7..
So two ways bouquets can be identical
1) All 21 and 91 together as 1 bouquet
2) We distribute 21 in 7 bouquets and 91 red too in 7 bouquets
So we will have 7 identical bouquets, each bouquet having 21/7=3 white and 91/7=13 red tulips..

So ans 7
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Here's my take.


Divide 91 red tulips by 21 white ones.
We get quotient of 4 and a remainder of 6 (these 6 are the white tulips) and 4 is the no. of identical bouquets. Now put the 6 white ones in each of the 2 bouquets (one white in first and one in second and so on). Thus we get 4+3 = 7

The part in paranthesis really got me confused here, and i could not understand anything written there. I get the part outside the paranthesis, but then... ?