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The letters A, B, C, D, E, and F represent digits and ABC,DEF represents a positive six-digit integer. What is the number ABC,DEF if 4(ABC,DEF) = 3(DEF,ABC)?

 Jan 29, 2024
 #1
avatar+261 
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\(400000a+40000b+4000c+400d+40e+4f=300000d+30000e+3000f+300a+30b+3c\)

\(3997(\overline{abc})=2996(\overline{def})\)

\(\overline{abcdef} = 29963997\)

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 Jan 30, 2024
 #2
avatar+259 
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Let's analyze the given information and solve for the six-digit number:

 

Equation: We are given the equation 4(ABC,DEF) = 3(DEF,ABC). This means that when we write the number ABC,DEF with the digits swapped (DEF,ABC), the result is 4/3 times the original number.

 

Digits and Positional Value: Each letter (A, B, C, D, E, F) represents a single digit. Since the number is positive and six-digit, the first digit (A) cannot be 0. Also, the digits cannot be repeated as the number is formed by swapping them later.

 

Analyzing the Equation: Let's break down the equation from the perspective of each digit's position:

 

Thousands digit: 4000A + 400B + 40C = 3000D + 300E + 30F

 

Hundreds digit: 400A + 40B + 40C = 300D + 30E + 30F

 

Tens digit: 40A + 4B + 4C = 30D + 3E + 3F

 

Units digit: 4A + 4B + 4C = 3D + 3E + 3F

 

Simplifying and Observing: We can notice that the equations for thousands and hundreds digits are identical. This means A + B + C = D + E + F. Also, the equations for tens and units digits are similar, with the coefficients differing. This suggests a relationship between the digits in these positions.

 

Trial and Error: Based on the observations, let's try assuming some values for the digits. Since A cannot be 0, let's start with A = 1. Then, to satisfy the equation for thousands and hundreds digits, we need D + E + F = 2. The only combination that works with these constraints and allows for different digits in the tens and units positions is D = 1, E = 0, and F = 1.

 

Solving for Remaining Digits: Now, we have A = 1, D = 1, E = 0, and F = 1. Substituting these values back into the equation for tens and units digits, we get:

4B + 4C = 3 + 3

4B + 4C = 6

 

This equation leaves us with two unknowns (B and C). However, we know that B and C cannot be the same due to the rule of non-repeating digits. Therefore, the only solution is B = 2 and C = 1.

 

Final Answer: By putting all the digits together, we obtain the six-digit number ABC,DEF = 121011. Therefore, the number that satisfies the given equation is 121011.

 Feb 6, 2024

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