Lesson: The GCD-LCM Formula

Comment on The GCD-LCM Formula

Hi Brent,

Can you please answer this question?

In a certain deck of cards, each card has a positive integer written on it. In a multiplication game, a child draws a card and multiplies the integer on the card by the next larger integer. If each possible product is between 15 and 200, then the least and greatest integers on the cards could be

(A) 3 and 15

(B) 3 and 20

(C) 4 and 13

(D) 4 and 14

(E) 5 and 14
gmat-admin's picture

Key: The question says "...each possible product is BETWEEN 15 and 200"
So, this means the products do no necessarily EQUAL 15 and 200

Let x be the number on the selected card.
So, x+1 = the next larger integer

Let's first test the LOWER LIMITS
We need (x)(x + 1) ≥ 15
Let's test some answer choices...

(A) 3 and 15
If x = 3, then (x + 1) = 4
Here, we get (x)(x + 1) = (3)(4) = 12
Since it is NOT the case that 12 ≥ 15, we know that the smallest number cannot be 3
ELIMINATE A and B

What about C?

(C) 4 and 13
If x = 4, then (x + 1) = 5
Here, we get (x)(x + 1) = (4)(5) = 20
Since it IS the case that 20 ≥ 15, we know that the smallest number must be 4
ELIMINATE E

Now let's test the UPPER LIMITS
We need (x)(x + 1) ≤ 20
(D) 4 and 14
If x = 14, then (x + 1) = 15
Here, we get (x)(x + 1) = (14)(15) = 210
Since it is NOT the case that 210 ≥ 200, we know that the biggest number cannot be 14
ELIMINATE D

By the process of elimination, the correct answer is C

Cheers,
Brent

Hey Brent,

I have a DS question and I do not understand the thinking behind this one.

If -5x = –x – 2 + 58x, and x does not equal zero, does x = -9?

(1) x does not equal 7
(2) x is less than zero

The answer is E, but I can not find an explanation.

Kind regards,
Niek
gmat-admin's picture

Hi Niek,

Sorry, but the equation in your question was spread out over about 10 lines.
I tried to reconstruct it (above), but it doesn't look right.
If you can tell be the equation, I'm sure I can help.

Cheers,
Brent

Hi Brent,

In your video lessons, you teach how to find LCM and GCF for given numbers, however there are a lot of problems that require to do the opposite, to find pairs of possible numbers for given LCM. Do you have a video explaining how to do that? You mention that we should master it, however, I find it hard to master something without a method. I see lots of people on GMAT forum easily suggest pairs for a given LCM, however I cannot understand how to come up with such pairs fast.
gmat-admin's picture

To find numbers that meet certain criteria (LCM or GCD), we need to keep in mind what each term means.

For example, if N is the GCD of integers J and K, we know that:
- N is a divisor of J and K
- In other words, N is "hiding" in the prime factorization of J and K
- N is the BIGGEST divisor of both J and K

So, for example, let's find two values (J and K) with a GCD of 15
- So 15 is "hiding" in the prime factorization of J and K
- Since 15 = 3 x 5, we know that a 3 and a 5 are "hiding" in the prime factorization of J and K

We can write:
J = (3)(5)(?)(?)(?)
K = (3)(5)(?)(?)(?)

At this point, we can see there are many possibilities....
For example, we could have:
J = (3)(5) = 15
K = (3)(5) = 15
J and K could both be 15

Or we could have:
J = (3)(5)(2) = 30
K = (3)(5) = 15

Or we could have:
J = (3)(5)(2)(7) = 210
K = (3)(5) = 15

Or we could have:
J = (3)(5)(2) = 30
K = (3)(5)(11) = 165

Or we could have:
J = (3)(5) = 15
K = (3)(5)(2)(2)(2) = 120

Or we could have:
J = (3)(5)(5) = 75
K = (3)(5)(2)(2)(2) = 120

In GENERAL, here are some quick and pairs of values that satisfy a given condition:
If N is the GCD of integers J and K, then some easy pairs of values include:
J = N and K = N
J = N and K = some multiple of N

-------------------------------

Now let's examine LCMs

For example, if N is the LCM of integers J and K, we know that:
- J and K are both divisors N
- J and K are both "hiding" in the prime factorization of N
- N is the SMALLEST multiple of both J and K

So, for example, let's find two values (J and K) with an LCM of 120
120 = (2)(2)(2)(3)(5)
So, J and K are both "hiding" in the product (2)(2)(2)(3)(5)

At this point, we can see there are many possibilities....
For example, we could have:
J = (2)(2)(2) = 8
K = (3)(5) = 15

Or we could have:
J = (2)(2)(2)(3)(5) = 120
K = (2)(2)(2)(3)(5) = 120

Or we could have:
J = (2)(2)(5) = 20
K = (2)(2)(2)(3) = 24

etc

In GENERAL, here are some quick and pairs of values that satisfy a given condition:
If N is the LCM of integers J and K, then some easy pairs of values include:
J = N and K = N
J = N and K = some divisor of N
----------------------------

Does that help?

Cheers,
Brent

https://gmatclub.com/forum/x-and-y-are-positive-integers-if-the-greatest-common-divisor-of-x-and-233373.html

For this question, why is it wrong to solve it this way . Given 3x and 9y LCM is 81, x =27 and y=9. I assume this is because we don't know for certain x =27 and y=9?
gmat-admin's picture

Question link: https://gmatclub.com/forum/x-and-y-are-positive-integers-if-the-greatest...

Your approach of choosing values is perfectly fine.
It's true that x = 27 and y = 9 satisfy the condition that the LCM of 3x and 9y is 81.
HOWEVER, the other condition is that the greatest common divisor (GCD) of x and 3y is 9
If x = 27 and y = 9, then the GCD of x and 3y is 27 (not 9)

Cheers,
Brent

I think Brent is the best math teacher, and the way he is there to help is admirable. Brent for president!
gmat-admin's picture

Ha! Thanks Dan!

https://gmatclub.com/forum/when-a-and-b-are-positive-integers-is-ab-a-multiple-of-278852.html

My brain pop up with this formula
(GCF OF AB)*(LCM OF AB)=AB.

Would you please explain this rule with the rule you did in the mentioned link?
gmat-admin's picture

Question link: https://gmatclub.com/forum/when-a-and-b-are-positive-integers-is-ab-a-mu...

When students see a Data Sufficiency question in which the two statements provide information about the GCD and LCM of two values respectively, many automatically assume that we'll need to apply the GCD-LCM formula. This isn't always the case, because there are various instances when we can answer the target question without having to apply the GCD-LCM formula.

In the linked question, I was able to apply a property that says:
If N is divisible by d, we can say that N = dk for some integer k
For example, if N is divisible by 5, we can say that N = 5k for some integer k

So, when statement 1 tells us that the greatest common divisor of A and B is 6, this is telling us that A and B are both divisible by 6.

So we can write A = 6k for some integer k, and B = 6j for some integer j

This means the product AB = (6k)(6j) = 36kj = (4)(9kj), so AB is definitely a multiple of 4.

Notice that, if the target question had asked us whether AB is a multiple of 8 (instead of a multiple of 4), statement 1 would NOT be sufficient, because we can't be certain that there's a 8 "hiding" in 36kj.

Does that help?

Pardon me I am posting same problem
Same problem but different method. Would you please say what i am missing?

https://gmatclub.com/forum/when-a-and-b-are-positive-integers-is-ab-a-multiple-of-278852.html
vs
https://www.beatthegmat.com/if-x-and-y-are-positive-integers-is-xy-a-multiple-of-8-t296815.html
gmat-admin's picture

Q1: https://gmatclub.com/forum/when-a-and-b-are-positive-integers-is-ab-a-mu...
Q2: https://www.beatthegmat.com/if-x-and-y-are-positive-integers-is-xy-a-mul...

Notice that Q2 has the exact same structure as Q1.
HOWEVER, Q1 asks whether AB is a multiple of 4, and Q2 asks whether xy is a multiple of 8.

Let's see what happens if we try to apply our Q1 strategy (in the comments above) to Q2.

In Q2, statement 1 tells us that the greatest common divisor of x and B is 10. This tells us that x and y are both divisible by 10.
So, we can write x = 10k for some integer k, and y = 10j for some integer j

This means the product xy = (10k)(10j) = 100kj
Can we be certain that 100kj is a multiple of 8? No.

We CAN show that 100kj is a multiple of 4, since 100kj = (4)(25kj)

However, we can't be certain that there's an 8 "hiding" in 100kj.
As such, we can't be certain that 100kj is a multiple of 8.

Does that help?

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