Question

When a 3 digit number 984 is added to another 3 digit number \(4 p 3\), we get a four digit number \(13 q 7\), which is divisible by 11 . The value of \(p+q\) is : (a) 10 (b) 11 (c) 12 (d) 13

Short answer

Answer: 10

Step-by-step solution

Identify the divisibility rule for 11

The divisibility rule for 11 states that a number is divisible by 11 if the difference between the sum of its alternate digits is either 0 or a multiple of 11.

Write the sum of the two numbers

When adding the two 3-digit numbers, they can be written as: \[ \begin{array}{@{}c@{\;}c@{}c@{}c@{}c} & & & 9 & 8 & 4 & \\ + & & & 4 & p & 3 & \\ \cline{1-6} & 1 & 3 & & q & 7 & \\ \end{array} \] From this, we have the following equation: \(984 + 4p3 = 13q7\)

Apply the divisibility rule for 11 and set up an equation

According to the divisibility rule for 11, the difference between the sum of the alternate digits in \(13q7\) must be a multiple of 11 or 0. Let's create an equation based on the rule: \((1 + q) - (3 + 7) = 11n\), where n is an integer. Simplifying the equation: \(1 + q - 10 = 11n\)

Simplify the equation

Simplify the equation further: \(q - 9 = 11n\) Now we need to find possible values of q such that the equation holds true.

Find possible values of q

Since q is a single digit number, the possible values for n should be small (probably either -1, 0, or 1). - If we assume n = 0: \(q - 9 = 0\) which means \(q = 9\) - If we assume n = 1: \(q - 9 = 11\) which means \(q = 20\). This is an invalid situation since q must be a single digit number. - If we assume n = -1: \(q - 9 = -11\) which means \(q = -2\). This is also an invalid situation since q must be a positive single digit number. So, the only valid value for q is 9.

Find the value of p

Now that we've found the value of q, let's substitute it into the equation for the addition of the three-digit numbers: \(984 + 4p3 = 1397\) Now we set up the following equation: \(984 + 400 + p0 + 3 = 1397\) Simplify the equation: \(p0 = 1397 - 984 - 400 - 3\) \(p0 = 10\) \(p = 1\)

Calculate the sum of p and q

Now that we have the values of p and q, we can add them together: \(p + q = 1 + 9 = 10\) The value of \(p+q\) is 10. Therefore, the correct answer is (a) 10.

Key concepts

Three-Digit Numbers

Three-digit numbers are the numbers that range from 100 to 999. Each number consists of three distinct places: the hundreds place, the tens place, and the units place. These places determine the overall value of the number. For example, in the number 984:

• The digit 9 is in the hundreds place, representing 900.
• The digit 8 is in the tens place, representing 80.
• The digit 4 is in the units place, representing 4.

When working with three-digit numbers, understanding their structure is vital for performing operations like addition (as seen in the exercise). It allows you to align digits correctly according to their place value, ensuring accurate calculations.
In the given exercise, two three-digit numbers are involved: 984 and 4p3, where 'p' is an unknown digit that must be found to solve the problem.

Addition of Numbers

Addition is a basic arithmetic operation where we combine values to get a total sum. In contexts involving multi-digit numbers, such as the three-digit numbers from our exercise, it's crucial to add the numbers column by column, starting from the rightmost digit.
Here's how the addition of 984 and 4p3 aligns:

• Start by adding the units digits: 4 + 3.
• Then proceed to the tens digits: 8 + p.
• Finally, add the hundreds digits: 9 and 4.

An important detail in this exercise is that the result should be a specific number, 13q7, which provides constraints and insights into what values 'p' and 'q' can be. This emphasizes the need for precision in aligning digits vertically to ensure accurate addition and the need to respect constraints imposed by the problem's requirements.

Single-Digit Constraints

In this problem, single-digit constraints play a significant role in narrowing down possible values for unknowns like 'p' and 'q' in the numbers provided. A single-digit number ranges from 0 to 9, making it important to check all possible values to meet conditions set by other factors, like divisibility rules.
For solving the problem:

• 'p' in 4p3 must make sense when added to 984 to become part of a valid four-digit number.
• 'q' in 13q7 has to satisfy a divisibility condition given by the divisibility rule for 11, which prescribes that the difference between the sum of odd and even positioned digits should be zero or a multiple of 11.

By exploring possible values under these constraints, the process allows us to determine suitable values for 'p' and 'q' that ensure all conditions are satisfied. In this scenario, 'p' equals 1, and 'q' equals 9, making the addition and divisibility rule work together harmoniously.