Math Problem Statement
Context: When 419 is added to the three-digit number 3p4, we get 7q3. If 7q3 is divisible by 3, what is the greatest possible value of p? p and q are digits.
Answer question
Solution
To solve this problem, let's break it down step by step using the given information and the rules for divisibility by 3.
Given Information:
- We have a three-digit number , where is a digit.
- When 419 is added to , the result is another three-digit number , where is a digit.
- The number is divisible by 3.
Step-by-Step Solution:
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Set up the equation:
The problem states that when 419 is added to , we get . Therefore, we can write the equation as:
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Convert the numbers into expressions:
Represent the numbers in a form suitable for calculations:
- can be expressed as .
- can be expressed as .
The equation becomes:
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Simplify the equation:
Simplify both sides:
Subtract 703 from both sides:
Divide the entire equation by 10:
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Condition for divisibility by 3:
For to be divisible by 3, the sum of its digits must be divisible by 3. Calculate the sum of the digits in :
For divisibility by 3:
Since , substitute in the divisibility condition:
For to be divisible by 3:
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Find the possible values of :
The number 12 is already divisible by 3. Therefore, must also be divisible by 3 for to be divisible by 3. The possible values for (as is a digit) are 0, 3, 6, and 9.
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Determine the greatest possible value of :
The greatest possible value of that is divisible by 3 is 9.
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Verify the solution with :
- If , then . But must be a digit, so cannot be 11.
- If , then .
With and , calculate as:
Check divisibility by 3:
18 is divisible by 3, so is divisible by 3.
Conclusion:
The greatest possible value of is 6.
Would you like more details or have any questions? Here are some related questions to further your understanding:
- What would be the value of if ?
- How does changing the digits of the number affect the divisibility of ?
- What is the smallest possible value of that makes divisible by 3?
- If was subtracted from instead of added to , how would the solution change?
- How many different valid pairs of exist such that is divisible by 3?
Tip: When checking for divisibility by a number, understanding the properties of that number can simplify the calculations.
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Math Problem Analysis
Mathematical Concepts
Equations
Divisibility
Number Theory
Formulas
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Theorems
Divisibility rule for 3
Suitable Grade Level
Grades 7-9