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GRE Prep Club Team Member
Joined: 20 Feb 2017
Posts: 2506
Given Kudos: 1055
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If x represents the sum of all the positive three-digit numbers that c
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04 Apr 2021, 22:51
04 Apr 2021, 22:51
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Question Stats:
45% (02:12) correct
54% (02:12) wrong based on 92 sessions
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If x represents the sum of all the positive three-digit numbers that can be constructed using each of the distinct nonzero digits a, b, and c exactly once, what is the largest integer by which x must be divisible?
(A) 3
(B) 6
(C) 11
(D) 22
(E) 222
(A) 3
(B) 6
(C) 11
(D) 22
(E) 222
If x represents the sum of all the positive three-digit numbers that c
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13 Oct 2021, 21:28
13 Oct 2021, 21:28
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If we consider the non-zero distinct numbers a,b,c - total 6 3-digit numbers can be constructed.
abc
acb
bca
bac
cba
cab
We can write abc as 100a + 10b +c. If we write every number in this format and add all these 6 numbers we get 200(a+b+c) + 20(a+b+c) + 2(a+b+c).
Now we take (a+b+c) as common and rewrite it as (a+b+c)(200 + 20 + 2) i.e (a+b+c)(222).
Hence it is conspicuous that 222 is the largest number we can divide with. So the ans is E
abc
acb
bca
bac
cba
cab
We can write abc as 100a + 10b +c. If we write every number in this format and add all these 6 numbers we get 200(a+b+c) + 20(a+b+c) + 2(a+b+c).
Now we take (a+b+c) as common and rewrite it as (a+b+c)(200 + 20 + 2) i.e (a+b+c)(222).
Hence it is conspicuous that 222 is the largest number we can divide with. So the ans is E
General Discussion
If x represents the sum of all the positive three-digit numbers that c
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06 Apr 2021, 10:35
06 Apr 2021, 10:35
4
Quote:
what is the largest integer by which x must be divisible
The answer choices are factors of x. Let's find the minimum value that x can have since the factor must be for any value of x.
Quote:
constructed using each of the distinct nonzero digits a, b, and c exactly once
Basically, there can be 6 such integers that can be constructed. To find the minimum sum of these integers, let's assume least values that a, b, and c can have: a=1, b=2, and c=3.
Next step is to arrange these digits, construct integers, and add to produce x.
1 2 3
1 3 2
2 1 3
2 3 1
3 1 2
3 2 1
Add up. x = 1332. This must be divisible by 3. Try dividing by 6, you get 222 as quotient. So, 222 must also divide x completely. We are looking for the largest divisor among the answer choices. Hence, E is the correct answer.
