If two integers p and q are in the ratio 4: 5 and their least common

Question

If two integers p and q are in the ratio 4:5 and their least common multiple is 800, what is their greatest common divisor?

A. 20
B. 40
C. 80
D. 100
E. 160

divyadna · Answer

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gmatophobia · Answer

LCM (p,q) =  2^2 * 2^3 * 5^2   (I have purposefully written  2^2  and  2^3  separately)

Given  \frac{p }{ q}  =  \frac{2^2 }{ 5}

This means, there was certain factor that was common both in p and q and that got cancelled out in the ratio.

Let's see if we can construct that factor -

LCM is obtained by considering the maximum power of the prime factors in p and q, therefore the missing  2^3  and 5 must be the common factor that got cancelled out when p was divided by q. Note that this factor is the greatest common factor, as any higher power of either 2 or of 5 in the common factor must be accounted in the LCM, and thus will change the value of the LCM.

Hence GCD =  2^3 * 5  = 40

We can cross verify  p = 2^2 * 2^3 * 5 ; q = 2^3 * 5^2

\frac{p}{q}  =  \frac{2^2 * 2^3 * 5 }{ 5 * 2^3 * 5 }

HCF of  p = 2^2 * 2^3 * 5 ; q = 2^3 * 5^2 = 40

Option B

Vivek1707 · Answer

HCF x LCM of two numbers = Product of two numbers

The numbers are 4x : 5x

Hence 20x^2 = 800y
So x^2 = 40y | Remember that 40y has to be perfect square
If y = 40, x = 40 and the numbers are 160 and 200 ---> LCM 800 matches
If y = 160, x would be 80 and the numbers would be 320 and 400 the LCM wont be 800

Hence answer is B

HuyBui · Answer

I’m not sure if this is a good way to solve this or not but I find the answer quite fast. 
Because the ratio is 4:5 and LCM is 800 so you can just find find the 2 number by 5a=4b=800, and you got a=160 and b=200 and HCF will be 40. 
I know this might not work with harder numbers but for these round numbers, I guess it’s quite simple and fast

egmat · Answer

Here's the  key idea : when two numbers are in a ratio, we can write them using a common multiplier.

Since p : q =  4  :  5 , let p =  4 k and q =  5 k, where k is some positive integer.

Now,  note that   4  and  5  share NO common factors (they're  coprime ). This means the ONLY common factor between  4 k and  5 k comes from the k part. So the GCD of p and q is simply k.

Next, let's find the LCM.  There's a beautiful formula:

LCM × GCD = p × q

Plugging in:
LCM × k =  4 k ×  5 k =  20 k2
LCM =  20 k

We're told LCM =  800 , so:
 20 k =  800 
k =  40

Since GCD = k, the GCD is  40 .

Answer: B

Quick verification:  p =  4  ×  40  =  160 , q =  5  ×  40  =  200 .
- GCD( 160 ,  200 ) =  40  ✓
- LCM( 160 ,  200 ) =  800  ✓

The general principle to remember:   If two numbers are in the ratio a : b where a and b are already in simplest form ( coprime ), then their GCD equals the multiplier k, and their LCM equals a × b × k. This relationship makes ratio + LCM/GCD problems very straightforward.

If two integers p and q are in the ratio 4: 5 and their least common

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