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# If there are four positive integers a, b, c and d in the ratio 1:2:3:4

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Math Expert
Joined: 02 Aug 2009
Posts: 7099
If there are four positive integers a, b, c and d in the ratio 1:2:3:4  [#permalink]

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22 Jul 2018, 03:01
10
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Difficulty:

95% (hard)

Question Stats:

41% (01:58) correct 59% (01:58) wrong based on 152 sessions

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If there are four positive integers a, b, c and d in the ratio 1:2:3:4 and their sum is a perfect square, what cannot be the value of any of the four number?
(A) 10
(B) 30
(C) 90
(D) 100
(E) 1000

New question

_________________

1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html
3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html

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Posts: 11
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Re: If there are four positive integers a, b, c and d in the ratio 1:2:3:4  [#permalink]

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22 Jul 2018, 03:08
chetan2u wrote:
If there are four positive integers a, b, c and d in the ratio 1:2:3:4 and their sum is a perfect square, what cannot be the value of any of the four number?
(A) 10
(B) 30
(C) 90
(D) 100
(E) 1000

New question
Is the answer is option (c) ? i.e. 9 ?

Thanks
Senior PS Moderator
Joined: 26 Feb 2016
Posts: 3325
Location: India
GPA: 3.12
If there are four positive integers a, b, c and d in the ratio 1:2:3:4  [#permalink]

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22 Jul 2018, 03:21
chetan2u wrote:
If there are four positive integers a, b, c and d in the ratio 1:2:3:4 and their sum is a perfect square, what cannot be the value of any of the four number?
(A) 10
(B) 30
(C) 90
(D) 100
(E) 1000

New question

The 4 positive numbers are in the ratio 1:2:3:4, the sum of which is 10x.

In order to eliminate answer options, let the first number(x) be one of the answer options.
The sum of the four positive integers will eliminate Option A as the sum is 100 which is $$10^2$$,
Option C as the sum is 900 which is $$30^2$$, and Option E as the sum if 10000 which is $$100^2$$.

To eliminate one answer option between Option B and Option D,
we can assume the number 30(Option C) as the third of the four numbers(of form 3x)
Now, the sum of the 4 numbers will be 10x = 100(which is $$10^2$$) as x = 10.

Therefore, Option D(100) is the only option that remains by elimination and is the correct answer!
_________________

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Math Expert
Joined: 02 Aug 2009
Posts: 7099
If there are four positive integers a, b, c and d in the ratio 1:2:3:4  [#permalink]

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22 Jul 2018, 03:30
2
archana89 wrote:
chetan2u wrote:
If there are four positive integers a, b, c and d in the ratio 1:2:3:4 and their sum is a perfect square, what cannot be the value of any of the four number?
(A) 10
(B) 30
(C) 90
(D) 100
(E) 1000

New question
Is the answer is option (c) ? i.e. 9 ?

Thanks

No, it is 90 itself..

Let the common ratio be X, so numbers are x:2x:3x:4x
And their sum = 10x
Now 10x is perfect square, so X has to be a multiple of 10*perfect square

A) 10 .... So 10*10....ok ... X is 10...10:20:30:40

B). 30 ..... 10*3 and we have a number that is a multiple of 3, so 30=3x, so X =10......10:20:30:40

C) 90 ...... 90=10*3^2, so X=90........90:180:270:360

D) 100... 10*10 and we do not have any number that has 10 in it...NOT possible

E). 1000.... 10*10^2....1000:2000:3000:4000

So D
_________________

1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html
3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html

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Joined: 31 May 2018
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Re: If there are four positive integers a, b, c and d in the ratio 1:2:3:4  [#permalink]

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22 Jul 2018, 03:33
1
chetan2u wrote:
If there are four positive integers a, b, c and d in the ratio 1:2:3:4 and their sum is a perfect square, what cannot be the value of any of the four number?
(A) 10
(B) 30
(C) 90
(D) 100
(E) 1000

New question

since a,b,c,d are in ratio 1:2:3:4
let a,b,c and d be x,2x,3x and 4x
x+2x+3x+4x=10x
10x is perfect square so \sqrt{10x^1/2} is integer
for \sqrt{10x^1/2} to be integer
x = 10,40,90,1000,......
value of any of four number can be
10,1000,90 and if x=10 then we can get 3x=30
so d) is the right answer
Intern
Joined: 18 Jun 2018
Posts: 11
Location: India
Concentration: Technology, Marketing
GPA: 3.98
Re: If there are four positive integers a, b, c and d in the ratio 1:2:3:4  [#permalink]

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22 Jul 2018, 09:09
1
chetan2u wrote:
archana89 wrote:
chetan2u wrote:
If there are four positive integers a, b, c and d in the ratio 1:2:3:4 and their sum is a perfect square, what cannot be the value of any of the four number?
(A) 10
(B) 30
(C) 90
(D) 100
(E) 1000

New question
Is the answer is option (c) ? i.e. 9 ?

Thanks

No, it is 90 itself..

Let the common ratio be X, so numbers are x:2x:3x:4x
And their sum = 10x
Now 10x is perfect square, so X has to be a multiple of 10*perfect square

A) 10 .... So 10*10....ok ... X is 10...10:20:30:40

B). 30 ..... 10*3 and we have a number that is a multiple of 3, so 30=3x, so X =10......10:20:30:40

C) 90 ...... 90=10*3^2, so X=90........90:180:270:360

D) 100... 10*10 and we do not have any number that has 10 in it...NOT possible

E). 1000.... 10*10^2....1000:2000:3000:4000

So D

Yes, Got it now ! Thanks for the explanantion !
VP
Joined: 07 Dec 2014
Posts: 1128
If there are four positive integers a, b, c and d in the ratio 1:2:3:4  [#permalink]

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22 Jul 2018, 10:12
chetan2u wrote:
If there are four positive integers a, b, c and d in the ratio 1:2:3:4 and their sum is a perfect square, what cannot be the value of any of the four number?
(A) 10
(B) 30
(C) 90
(D) 100
(E) 1000

sum of sequence=10a, so a=1/10 of sum, or perfect square
10 is first term of sequence summing to 100, or 10^2 NO
30 is third term of sequence summing to 100, or 10^2 N0
90 is first term of sequence summing to 900, or 30^2 N0
1000 is first term of sequence summing to 10,000, or 100^2 N0
by default, 100 cannot be the value of any of the four numbers
D
VP
Joined: 09 Mar 2016
Posts: 1208
If there are four positive integers a, b, c and d in the ratio 1:2:3:4  [#permalink]

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27 Jul 2018, 05:42
pushpitkc wrote:
chetan2u wrote:
If there are four positive integers a, b, c and d in the ratio 1:2:3:4 and their sum is a perfect square, what cannot be the value of any of the four number?
(A) 10
(B) 30
(C) 90
(D) 100
(E) 1000

New question

The 4 positive numbers are in the ratio 1:2:3:4, the sum of which is 10x.

In order to eliminate answer options, let the first number(x) be one of the answer options.
The sum of the four positive integers will eliminate Option A as the sum is 100 which is $$10^2$$,
Option C as the sum is 900 which is $$30^2$$, and Option E as the sum if 10000 which is $$100^2$$.

To eliminate one answer option between Option B and Option D,
we can assume the number 30(Option C) as the third of the four numbers(of form 3x)
Now, the sum of the 4 numbers will be 10x = 100(which is $$10^2$$) as x = 10.

Therefore, Option D(100) is the only option that remains by elimination and is the correct answer!

pushpitkc please reveal a flaw in my reasoning below

i found two correct answers to this question ! friends would you please share your sharp math knowledge
.....
(A) 10 ( 10+20+30+40 = 100 (perfetct square)

(B) 30 ( 30+60+90+120 = 300 ) (its not a perfetct square )

(C) 90 ( 90+180+270+360 (perfetct square)

(D) 100 (100+200+300+400 = 1000 (its not a perfetct square)

(E) 1000 (1000+2000+3000+4000) = 10,000 (perfetct square)
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Joined: 26 Feb 2016
Posts: 3325
Location: India
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If there are four positive integers a, b, c and d in the ratio 1:2:3:4  [#permalink]

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29 Jul 2018, 21:07
1
dave13 wrote:
pushpitkc wrote:
chetan2u wrote:
If there are four positive integers a, b, c and d in the ratio 1:2:3:4 and their sum is a perfect square, what cannot be the value of any of the four number?
(A) 10
(B) 30
(C) 90
(D) 100
(E) 1000

New question

The 4 positive numbers are in the ratio 1:2:3:4, the sum of which is 10x.

In order to eliminate answer options, let the first number(x) be one of the answer options.
The sum of the four positive integers will eliminate Option A as the sum is 100 which is $$10^2$$,
Option C as the sum is 900 which is $$30^2$$, and Option E as the sum if 10000 which is $$100^2$$.

To eliminate one answer option between Option B and Option D,
we can assume the number 30(Option C) as the third of the four numbers(of form 3x)
Now, the sum of the 4 numbers will be 10x = 100(which is $$10^2$$) as x = 10.

Therefore, Option D(100) is the only option that remains by elimination and is the correct answer!

pushpitkc please reveal a flaw in my reasoning below

i found two correct answers to this question ! friends would you please share your sharp math knowledge
.....
(A) 10 ( 10+20+30+40 = 100 (perfetct square)

(B) 30 ( 30+60+90+120 = 300 ) (its not a perfetct square )

(C) 90 ( 90+180+270+360 (perfetct square)

(D) 100 (100+200+300+400 = 1000 (its not a perfetct square)

(E) 1000 (1000+2000+3000+4000) = 10,000 (perfetct square)

Hey dave13

You are absolutely correct with your astute observation

As you have rightly pointed out when you have 30 and 100 as the first number of the sequence,
you will come up with a sum (10x) which is not a perfect square. However, in order to eliminate
one of two correct answer options, you can use 30 as the third number(3x) making Option B, the
sum of four number a perfect square, 100(10 + 20 + 30 + 40)

That is why Option D is the correct answer!

_________________

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If there are four positive integers a, b, c and d in the ratio 1:2:3:4 &nbs [#permalink] 29 Jul 2018, 21:07
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