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What is the smallest possible sum of nonnegative integers a, b, anD

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Bunuel
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What is the smallest possible sum of nonnegative integers a, b, anD [#permalink] New post   21 Mar 2017, 05:06
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What is the smallest possible sum of nonnegative integers a, b, and c such that 36a + 6b + c = 173?

A. 5
B. 7
C. 13
D. 16
E. 18
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C
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BrentGMATPrepNow
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Re: What is the smallest possible sum of nonnegative integers a, b, anD [#permalink] New post   21 Mar 2017, 08:11
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Bunuel wrote:
What is the smallest possible sum of nonnegative integers a, b, and c such that 36a + 6b + c = 173?

A. 5
B. 7
C. 13
D. 16
E. 18


First off, we can minimize the sum of a, b and c by maximizing the value of a.
(36)(5) = 180, so 5 is too big
(36)(4) = 144
So, a = 4

173 - 144 = 29.
So, we now have 6b + c = 29
We can minimize the sum of b and c by maximizing the value of b.
(6)(5) = 30, so 5 is too big
(6)(4) = 24
So, b = 4

If a = 4 and b = 4, then 36a + 6b + c = 144 + 24 + c = 173
So, c = 5

So, the minimum value of a + b + c is 4 + 4 + 5 = 13
Answer:
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C


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mesutthefail
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Re: What is the smallest possible sum of nonnegative integers a, b, anD [#permalink] New post   21 Mar 2017, 08:31
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To make a,b and c as small as possible, we should maximize the value with greatest multiplied( 36a), and do the same operation on the remaining parts afterwards.

36= 36x5=180 so , 36x4 = 144, a =4

This leaves 6b+c = 29, ====> 6x5=30 ; 6x4= 24, b =4

c=5, therefore 4+4+5= 13.

C
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Re: What is the smallest possible sum of nonnegative integers a, b, anD [#permalink] New post   21 Mar 2017, 22:54
Bunuel wrote:
What is the smallest possible sum of nonnegative integers a, b, and c such that 36a + 6b + c = 173?

A. 5
B. 7
C. 13
D. 16
E. 18


OFFICIAL SOLUTION



In order to minimize the total value of a, b, and c, we should make a as large as possible -- since we'll simply reach 173 much faster moving in chunks of 36 than we would by moving in chunks of 6 or 1.

Since 173÷36=4 remainder 29, we will choose a=4 and move on to consideration of b and c.

In the same manner, we will now maximize b, since moving in chunks of 6 is much more efficient than moving in chunks of 1.
Since 29÷6=4 remainder 5, we will choose b=4. At this point we will be left to choose c=5 to cover the remaining value.

Thus the minimum possible value of a+b+c is 4+4+5=13, and answer C is correct.
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Re: What is the smallest possible sum of nonnegative integers a, b, anD [#permalink] New post   22 Mar 2017, 13:56
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My approach:
Write a list of all of the multiples of:

36 - 36, 72, 108, 144
6 - 6, 12, 18, 24, 30, 36, 42, 48
1 - 1, 2, 3, 4, 5

Look at the numbers for a while.

You would observe that 144 + 24 + 5 = 173

Therefore a is 4, b is 4 and c is 5

Add up all numbers together 4 + 4 + 5 = 13

Which is answer C
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Re: What is the smallest possible sum of nonnegative integers a, b, anD [#permalink] New post   18 May 2023, 10:22
My approach :
6(6a+b)+c = 173 => the nearest multiple of 6 for the number 173 is 168
therefore the expression can be expressed as 6[6{4}+4]+5 => a=4, b=4 and c=5. these are the least values of a,b and c that satisfy the equation.
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Re: What is the smallest possible sum of nonnegative integers a, b, anD [#permalink]
18 May 2023, 10:22
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