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# If a, b and c are all positive integers , and 2a=7b=8c , then what is

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If a, b and c are all positive integers , and 2a=7b=8c , then what is  [#permalink]

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03 Aug 2017, 03:24
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87% (01:46) correct 13% (02:12) wrong based on 85 sessions

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If a, b and c are all positive integers, and 2a=7b=8c, then what is the LEAST possible value of a+b+c?

A. 26
B. 34
C. 43
D. 50
E. 86

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Re: If a, b and c are all positive integers , and 2a=7b=8c , then what is  [#permalink]

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03 Aug 2017, 04:28
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1
Skywalker18 wrote:
If a, b and c are all positive integers, and 2a=7b=8c, then what is the LEAST possible value of a+b+c?

A. 26
B. 34
C. 43
D. 50
E. 86

When I see two "=", I automatically tend to put another one.

2a = 7b = 8c = k

a = k/2
b = k/7
c = k/8

a:b:c = k/2 : k/7 : k/8
Multiply the ratio by 56 to get
a:b:c = 28k:8k:7k

If k = 1 (since a, b, c are positive integers), a = 28, b = 8 and c = 7 (the minimum possible values)
So min sum = 28 + 8 + 7 = 43

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Karishma
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Re: If a, b and c are all positive integers , and 2a=7b=8c , then what is  [#permalink]

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03 Aug 2017, 04:08
Since 7 and 2 are primes.
C will need to be 7 , which makes 8c =56 and B to be 8 for 7a to be equal to 8c .
In the similar fashion a will be 28. So adding all the terms you get 43.

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Re: If a, b and c are all positive integers , and 2a=7b=8c , then what is  [#permalink]

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03 Aug 2017, 04:31
Skywalker18 wrote:
If a, b and c are all positive integers, and 2a=7b=8c, then what is the LEAST possible value of a+b+c?

A. 26
B. 34
C. 43
D. 50
E. 86

I would go for C

2A =7b=8C

Min value (Used case method/trial method)
2(4*7)=7 (8) =8(7)

28+8+7 =43
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Re: If a, b and c are all positive integers , and 2a=7b=8c , then what is  [#permalink]

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03 Aug 2017, 04:39
Yes 43, makes sense to find a factor for the three

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Re: If a, b and c are all positive integers , and 2a=7b=8c , then what is  [#permalink]

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16 Oct 2018, 08:08
Skywalker18 wrote:
If a, b and c are all positive integers, and 2a=7b=8c, then what is the LEAST possible value of a+b+c?

A. 26
B. 34
C. 43
D. 50
E. 86

When I see two "=", I automatically tend to put another one.

2a = 7b = 8c = k

a = k/2
b = k/7
c = k/8

a:b:c = k/2 : k/7 : k/8
Multiply the ratio by 56 to get
a:b:c = 28k:8k:7k

If k = 1 (since a, b, c are positive integers), a = 28, b = 8 and c = 7 (the minimum possible values)
So min sum = 28 + 8 + 7 = 43

Nice one,enjoy how Karishma solves her problems, keep on the good work
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Re: If a, b and c are all positive integers , and 2a=7b=8c , then what is  [#permalink]

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17 Oct 2018, 19:04
Skywalker18 wrote:
If a, b and c are all positive integers, and 2a=7b=8c, then what is the LEAST possible value of a+b+c?

A. 26
B. 34
C. 43
D. 50
E. 86

The LCM of 2, 7, and 8 is 56. Thus, we let 8c = 56, which yields c = 7. If c = 7, then b = 8, and a = 28, so the least value of the sum of a, b, and c is 43.

Alternate Solution:

Let’s focus on the equation 7b = 8c. We notice that 7 divides 8c (since 8c equals some integer times 7), but 7 does not divide 8; therefore 7 must divide c. Then, the smallest choice for c is 7. Similarly, 8 divides 7b and since 8 does not divide 7, it must divide b; therefore the smallest choice for b is 8. Since 2a = 8c = 8(7) = 56, a = 28. Thus, the smallest value of a + b + c is 28 + 8 + 7 = 43.

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Re: If a, b and c are all positive integers , and 2a=7b=8c , then what is   [#permalink] 17 Oct 2018, 19:04
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