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# The product of the GCF and LCM of integers p and q is rs. What is the

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Joined: 07 Jun 2018
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The product of the GCF and LCM of integers p and q is rs. What is the  [#permalink]

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25 Oct 2018, 12:05
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25% (medium)

Question Stats:

74% (01:08) correct 26% (00:40) wrong based on 35 sessions

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The product of the GCF and LCM of integers p and q is rs. What is the LCM of p and q?
1. $$r=\frac{400}{s}$$
2. The GCF of p and q is 20
GMATH Teacher
Status: GMATH founder
Joined: 12 Oct 2010
Posts: 485
Re: The product of the GCF and LCM of integers p and q is rs. What is the  [#permalink]

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25 Oct 2018, 14:25
Jazzmin wrote:
The product of the GCF and LCM of positive integers p and q is rs. What is the LCM of p and q?
1. $$r=\frac{400}{s}$$
2. The GCF of p and q is 20

$$\left\{ \matrix{ p,q\,\, \ge \,\,1\,\,\,{\rm{ints}} \hfill \cr pq = GCF\left( {p,q} \right) \cdot LCM\left( {p,q} \right) = rs\,\,\,\left( * \right) \hfill \cr} \right.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,? = LCM\left( {p,q} \right)$$

$$\left( 1 \right)\,\,\,rs = 400\,\,\,\left\{ \matrix{ \,{\rm{Take}}\,\,\left( {p,q} \right) = \left( {1,400} \right)\,\,\,\, \Rightarrow \,\,\,{\rm{?}}\,\,{\rm{ = }}\,\,{\rm{400}}\,\, \hfill \cr \,{\rm{Take}}\,\,\left( {p,q} \right) = \left( {20,20} \right)\,\,\,\, \Rightarrow \,\,\,{\rm{?}}\,\,{\rm{ = }}\,\,{\rm{20}}\,\,\, \hfill \cr} \right.$$

$$\left( 2 \right)\,\,GCF\left( {p,q} \right) = 20\,\,\,\left\{ \matrix{ \,\left( {{\mathop{\rm Re}\nolimits} } \right){\rm{Take}}\,\,\left( {p,q} \right) = \left( {20,20} \right)\,\,\,\, \Rightarrow \,\,\,{\rm{?}}\,\,{\rm{ = }}\,\,{\rm{20}}\,\, \hfill \cr \,{\rm{Take}}\,\,\left( {p,q} \right) = \left( {20,40} \right)\,\,\,\, \Rightarrow \,\,\,{\rm{?}}\,\,{\rm{ = }}\,\,{\rm{40}}\,\,\, \hfill \cr} \right.\,$$

$$\left( {1 + 2} \right)\,\,\,\,\,\left( * \right)\,\,\,\, \Rightarrow \,\,\,\,20 \cdot LCM\left( {p,q} \right) = 400\,\,\,\,\, \Rightarrow \,\,\,\,\,\,? = LCM\left( {p,q} \right)\,\,{\rm{ = }}\,\,{\rm{20}}\,\,\,\,\, \Rightarrow \,\,\,\,\,\,{\rm{SUFF}}.$$

This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.
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Re: The product of the GCF and LCM of integers p and q is rs. What is the  [#permalink]

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25 Oct 2018, 14:53
Jazzmin wrote:
The product of the GCF and LCM of integers p and q is rs. What is the LCM of p and q?
1. $$r=\frac{400}{s}$$
2. The GCF of p and q is 20

Product of LCM(p,q) and GCF (p,q) = p*q

We are given LCM (p,q) * GCF (p,q) = rs

rs = pq

To find the LCM of (p,q) we need the GCF (p,q) and the product of p and q

Statement 1) tells us rs = 400 so pq = 400

Not enough we need another variable.

Insufficient.

Statement 2) GCF (p,q) = 20

Not enough we don’t have p * q

Combined

We have both we can find the LCM.

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The product of the GCF and LCM of integers p and q is rs. What is the  [#permalink]

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25 Oct 2018, 22:27
Hi,

This question tests the basic rule of L.C.M and H.C.F

Rule:

If “x” and “y” are two numbers, then

Product of two numbers = Product of their L.C.M and H.C.F

i.e.,

(x*y) = (L.C.M(x,y) * H.C.F(x,y)).

Given in this question,

Two numbers are “p” and “q”.

GCF and LCM product is r*s

According to the above rule,

p * q = r*s

Question:

LCM of (p,q) ?

Statement I is insufficient:

r = 400/s

From the rule we can understand that, r*s = p*q

So, r*s = 400

i.e., p*q = 400.

Not enough find out the L.C.M

If p = 10 and q = 40

Then L.C.M is 40 and G.C.F is 10

But if p = 25 and q = 16

Then L.C.M is 400 and G.C.F is 1.

Different values of L.C.M.

So not sufficient.

Statement II is insufficient:

The GCF of p and q is 20

Knowing only G.C.F doesn’t help, because we don’t know the two numbers “p” and “q”

So not sufficient.

Together it is sufficient.

We know the product of two numbers and we know the G.C.F is 20,

So L.C.M is also 20.

So sufficient.

Together it is sufficient.

Hope this helps.
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The product of the GCF and LCM of integers p and q is rs. What is the &nbs [#permalink] 25 Oct 2018, 22:27
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