Finding Unknown Digits Using Divisibility Rules

We are given a 10-digit number, M30348462N, which is known to be divisible by both 8 and 11. Our goal is to find the values of the digits M and N and then calculate the expression \(M^2 + N^2 - 18\).

Since M is the first digit of a 10-digit number, M must be a digit from 1 to 9. N is the last digit, so it can be any digit from 0 to 9.

Applying Divisibility Rule for 8

A number is divisible by 8 if the number formed by its last three digits is divisible by 8. In the number M30348462N, the last three digits form the number 62N.

So, the number 62N must be divisible by 8. We can test the possible values for N from 0 to 9 to see which one makes 62N divisible by 8.

• If N = 0, 620 is not divisible by 8 (\(620 \div 8 = 77.5\)).
• If N = 1, 621 is not divisible by 8.
• If N = 2, 622 is not divisible by 8.
• If N = 3, 623 is not divisible by 8.
• If N = 4, 624 is divisible by 8 (\(624 \div 8 = 78\)). This is a possible value for N.
• If N = 5, 625 is not divisible by 8.
• If N = 6, 626 is not divisible by 8.
• If N = 7, 627 is not divisible by 8.
• If N = 8, 628 is not divisible by 8 (\(628 \div 8 = 78.5\)).
• If N = 9, 629 is not divisible by 8.

From the above checks, the only digit N that makes 62N divisible by 8 is N = 4.

So, we have found that N = 4.

Applying Divisibility Rule for 11

A number is divisible by 11 if the alternating sum of its digits, starting from the rightmost digit (N), is divisible by 11.

The digits of the number M30348462N from right to left are N, 2, 6, 4, 8, 4, 3, 0, 3, M.

The alternating sum is calculated as:

\(N - 2 + 6 - 4 + 8 - 4 + 3 - 0 + 3 - M\)

Substitute the value of N = 4 that we found:

\(4 - 2 + 6 - 4 + 8 - 4 + 3 - 0 + 3 - M\)

Let's simplify the sum:

\((4 - 2) + (6 - 4) + (8 - 4) + (3 - 0) + 3 - M\)

\(2 + 2 + 4 + 3 + 3 - M\)

\(4 + 4 + 6 - M\)

\(8 + 6 - M\)

\(14 - M\)

According to the divisibility rule for 11, this alternating sum \(14 - M\) must be a multiple of 11. Possible multiples of 11 are ..., -22, -11, 0, 11, 22, ...

We know that M is a single digit from 1 to 9. Let's check which multiple of 11 makes sense:

• If \(14 - M = 0\), then \(M = 14\). This is not a single digit.
• If \(14 - M = 11\), then \(M = 14 - 11 = 3\). This is a valid digit (1-9).
• If \(14 - M = 22\), then \(M = 14 - 22 = -8\). This is not a valid digit.
• If \(14 - M = -11\), then \(M = 14 + 11 = 25\). This is not a single digit.

The only possible single digit value for M from 1 to 9 that makes \(14 - M\) a multiple of 11 is M = 3.

So, we have found that M = 3.

Verifying the Number

With M=3 and N=4, the number is 3303484624.

• Last three digits are 624, which is divisible by 8 (\(624 \div 8 = 78\)).
• Alternating sum of digits: \(4 - 2 + 6 - 4 + 8 - 4 + 3 - 0 + 3 - 3 = 11\), which is divisible by 11.

The values M=3 and N=4 satisfy both divisibility conditions.

Calculating \(M^2 + N^2 - 18\)

Now we need to find the value of the expression \(M^2 + N^2 - 18\) using M = 3 and N = 4.

Substitute the values:

\(M^2 + N^2 - 18 = 3^2 + 4^2 - 18\)

Calculate the squares:

\(3^2 = 3 \times 3 = 9\)

\(4^2 = 4 \times 4 = 16\)

Now substitute these values back into the expression:

\(9 + 16 - 18\)

Perform the addition and subtraction:

\(25 - 18\)

\(7\)

The value of \(M^2 + N^2 - 18\) is 7.

Revision Table: Key Divisibility Rules

Additional Information on Number Theory Problems

Problems involving finding unknown digits in a number based on divisibility rules are common in number theory and quantitative aptitude sections of exams. Understanding and quickly applying these rules is crucial.

When solving such problems:

• Start with the divisibility rule that gives you the most direct information about the unknown digits, usually the last digits (like rules for 2, 5, 10, 4, 8, 25, 125).
• Then, apply rules that involve the sum or alternating sum of digits (like rules for 3, 9, 11) to find other unknown digits.
• Always verify your found digits by checking if the resulting number satisfies all given conditions.

Mastering these divisibility rules helps in solving a variety of problems efficiently. Practice applying them to different numbers and scenarios.