a is the hundreds digit of the three digit integer x, b

As this question is written, I was confused by the portion "the different between the two greatest possible values of x"?
Isn´t the word "greatest" redundant?

Aren´t only two possible values for x, namely 842 and 421?

Hi all,

Let me show how I did it - and I am quite happy as I HATE number properties questions ...

First of all we have to start somewhere. And since this is a 3-digit number then it must be 100<x<999.

We also know that a=2b=2c. So, c will be 1/4 of a and b will be 1/2 of a. We are interested in a then!

In 999, the last digit is 9. So, a can never be 4 times 9, because it would be 36, making b 18. This is not a 3-digit number. In other words, we are looking for 2s to divide and multiply.

So, 8 is the greatest multiple of 2, from 1 to 9.
If a=8, then b=4 and c=2 --> x = 842

We cannot have a=2, as then c would be 1/2.

So, the lowest possibe a=4, making b=2 and c=1 --> x = 421.

Finally, 842 - 421 = 421, ANS D

Ratio of a:b:c= 4:2:1

Two possible greatest single digit values for a are 8 and 4

if a is 8, then x= 842
if a is 4, then x= 421

difference= 842-421= 421

D is the answer

Just to add to this solution =>
I think There are only two possible values of x here they are => 842 and 421

Frame the number and assume a(hundred's) b(ten's) c(unit's) place in terms of c as 4c 2c c.
Now substitute c=2 once and c=1.
The numbers obtained in each iteration (842 -421) using the above values gives 421.
Option D.

Our 3-digit number looks like this: abc

If a = 4c, then c cannot be greater than 2. Otherwise, a will not be a digit.
For example, if we let c = 3, then a = 12, and 12 is not a digit.

So, there are ONLY TWO possible values for c.
Case a: c = 1
If a = 2b = 4c, the a = 4, b = 2 and c = 1
So, the resulting 3-digit number (aka x) is 421

Case b: c = 2
If a = 2b = 4c, the a = 8, b = 4 and c = 2
So, the resulting 3-digit number (aka x) is 842

What is the difference between the two greatest possible values of x?
There are only 2 possible values of x.
Their DIFFERENCE = 842 - 421 = 421

Answer: D

Cheers,
Brent

We see that the value of c is the smallest. Since a > 0, b and c are also greater than 0; therefore,

If c = 1, we have a = 4, b = 2, and hence, x = 421.

If c = 2, we have a = 8, b = 4, and hence, x = 842.

Since c can’t be 3 or more (otherwise a would be a two-digit number), there are only 2 possible values of x, and their difference is 842 - 421 = 421.

Answer: D

We see that the value of c is the smallest. Since a > 0, b and c are also greater than 0; therefore,

If c = 1, we have a = 4, b = 2, and hence x = 421.

If c = 2, we have a = 8, b = 4, and hence x = 842.

Since c can’t be 3 or more (otherwise a would be a two-digit number), there are only 2 possible values of x, and their difference is 842 - 421 = 421.

Answer: D

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