Question

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

• A. 56
• B. 59
• C. 49
• D. 46

Answer 1

The Correct Option is D

Given the equations: \( ab = 432 \), \( bc = 96 \), and \( c < 9 \). We need to determine the smallest possible value of \( a + b + c \).

First, express \( a \) in terms of \( b \) and \( c \):

\( a = \frac{432}{b} \) 

\( b = \frac{96}{c} \)

Substituting \( b \) from the second equation into the first:

\( a = \frac{432}{\frac{96}{c}} = \frac{432c}{96} = \frac{9c}{2} \)

Since \( a \) must be an integer, \( \frac{9c}{2} \) must be an integer:

This implies \( c \) must be even. Given \( c < 9 \), the possible values for \( c \) are 2, 4, 6, and 8.

Calculate \( a + b + c \) for valid \( c \) values:

c	a	b	a+b+c
2	\( \frac{9 \times 2}{2} = 9 \)	\( \frac{96}{2} = 48 \)	9 + 48 + 2 = 59
4	\( \frac{9 \times 4}{2} = 18 \)	\( \frac{96}{4} = 24 \)	18 + 24 + 4 = 46
6	\( \frac{9 \times 6}{2} = 27 \)	\( \frac{96}{6} = 16 \)	27 + 16 + 6 = 49
8	\( \frac{9 \times 8}{2} = 36 \)	\( \frac{96}{8} = 12 \)	36 + 12 + 8 = 56

The smallest possible value of \( a + b + c \) is 46 when \( c = 4 \).

Answer 2

\( a \times b = 432 \) 
\( b \times c = 96 \) 
with the constraint \( c < 9 \)

Step 1: Factor Pairs of \( b \times c = 96 \)

We list all factor pairs such that \( c < 9 \):

• \( 48 \times 2 \)
• \( 32 \times 3 \)
• \( 24 \times 4 \)
• \( 16 \times 6 \)
• \( 12 \times 8 \)

Step 2: Factor Pairs of \( a \times b = 432 \)

We consider each value of \( b \) from above and check if \( a = \frac{432}{b} \) is an integer:

• For \( b = 24 \Rightarrow a = \frac{432}{24} = 18 \)
• Also, \( b = 24 \Rightarrow c = \frac{96}{24} = 4 \)

Step 3: Compute the Minimum Sum

\( a + b + c = 18 + 24 + 4 = 46 \)

 Final Answer:

The minimum value of \( a + b + c \) is 46. 
Correct Option: (D)