If a, b and c are positive integers such that ab = 432, bc = 96 and c < 9, then the smallest possible value of a + b + c is
- 56
- 59
- 49
- 46
The Correct Option is D
Approach Solution - 1
The correct answer is (D): \(46\)
Given \(ab = 432\), \(bc = 96\) and \(c < 9\)
To find the minimum value for \(a + b + c\), the two larger numbers should be as close as possible.
The closest combination whose product is \(432\) is \(18 × 24\).
For \(b = 24\), we get \(c = 4\) and \(a = 18\).
Hence the least value for \(a + b + c = 46\)
Approach Solution -2
Given that our aim is to minimize the sum, we can have the following possible combinations for \( b× c = 96\) since \(c<9: \)
\(48× 2 ; 32 × 3 ; 24 × 4 ; 16×6 ; 12 × 8 \)
In a similar manner, we can factorize \(𝑎 × 𝑏 = 432\) into its component parts. Upon careful examination, we find that \(18 × 24\) 𝑎 𝑛 𝑑 \(24 × 4\), which correspond to \(𝑎 × 𝑏\) and \(b× c\), respectively, together give us the least value of the sum of
\(𝑎 + 𝑏 + 𝑐 = 18 + 24 + 4 = 46. \)
Therefore, Option D is the right response.
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