Modulus Pattern for Divisibility

There are a lot of divisibility rules floating around us. We knew exactly when a number is divisible by a certain divisor:

1. A number is divisible by 2 if and only if it ends in 0,2,4,6, and 8.
2. A number is divisible by 3 if and only if the digits of the number add up to a multiple of 3.
3. A number is divisible by 4 if and only if the last two digits are divisible by 4.
4. A number is divisible by 5 if and only if its last digit is either 5 or 0.

Also, we learned the other divisibility rules in large numbers through combining divisibility rules:

1. A number is divisible by 14 if and only if it is both divisible by 7 and 2.
2. A number is divisible by 100 if and only if the last two digits are both zeroes (both divisible by 25 and 4).
3. A number is divisible by 143 if and only if it is both divisible by 11 and 13.
4. A number is divisible by 42 if and only if it is both divisible by 6 and 7.

Example Question 1

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This is the answer to the question, with a detailed solution. If math is needed, it can be done inline: \( x^2 = 144 \), or it can be in a centered display:

\[ \frac{x^2}{x+3} = 4y \]

And our final answer is 10. \( _\square \)