Arithmetic
1 Problem • 3 sub-topics
Adalynn Le • 5/3/2026
Introduction
If you're taking the AMC 10, chances are you know how to do arithmetic, even if you don't kow what it is. Arithmetic is the most basic type of math, and it involves using the four basic operations (addition, subtraction, multiplication, and division) to compute expressions.
Basic Operations
This is a pretty simple subject, so let's just go over the basic tools of it
Addition - Finding the total when combining two quantities
Subtraction - Finding the difference or quantity between the values of two quantities
Multiplication - Repeated addition in an array
Division - Repeated subtraction
Now don't fall into the trap of thinking there's nothing to learn in Arithmetic, or that Arithmetic is confined to only those subjects. There are more operations more things to do, based on the four basic ones. In fact, technically addition could be called the only main operation, because subtraction is negative addition, multiplication is repeated addition, and division is repeated subtraction. The point of this essentially, is that arithmeitc is a complex system made from a basic building blocks. Let's go over some bigger operations in arithmetic.
Powers
Powers are to multiplication what multiplication is to addition. It's repeated multiplication. The inverse is is radicals and roots.
Series and Sequences
A sequence is effectively what happens when you list each value in the process of performing a power or multiplication. An arithmetic series is when you continously add, (multiplication) and you take each sum. The same is true for geometric series and multiplication. The series is the sum of the values in a sequence.
Factoring and Division
I want to dedicate this section to division particularly, specifically because I think there's so much you can do with it. division is, in my eyes, the most versatile operation because it unveils a lot about number theory, especially when you get into factorization.
Different Representations of Division
Obviously, as an operation, there exists just \(\div\), but division in meaning can be split into different ways: arithmetic expressions, fractions, and ratios.
Numerator: \(1\)
Denominator: \(1\)
Arithmetic Representation
\(1 \div 1 = 1\)
Fraction Representation
\(\frac{1}{1}\)
Ratio Representation
\(1:1\)
GCF and LCM
This is a bit of a side tangent, because neither of these metrics relate directly to arithmetic, but I want to illustrate the applicability and importance of division in other systems. GCF is the largest value such that two or more integers are evenly divisible by it. LCM, on the other hand, details the smallest value that has two or more integers are factors. Knowing these can help you save a lot of time when doing arithmetic. For example, when calculating with fractions, it allows you to change the denominator or simplify quickly. You can find both of these metrics through prime factorization.
Prime Factorization
Prime factorization is what it sounds like, divindg continously until you reduce a number into it's prime factors. An easy way to do this is with the use of a factor tree, but other methods exist. A prime factorization is helpful because the GCF of two numbers is always the product of the overlapping factors and the LCM is the product of one number with the other number divided by the GCF. The prime factorization can also let you find the total number of factors by raising each power by one and multiplying them all together.
Quick Math
Let's be honest. We all know how to do arithmetic, but knowing it doesn't make you good. Especially on timed tests like the AMC 10, every second counts, you want to get things done fast and accurately. This can also just help you function in real-world settings and allow you to be more focused and successful
Divisiliby Rules
Divisibility rules are quick ways to tell if a number is divisible by a number. The most common ones are for \(2, 3\) and \(5\), which pretty much everybody knows (being even, having a sum of digits of divisible by \(3\), and ending in \(0\) or \(5\), respectively), but there are a lot more. For powers of \(2\) represented as \(2^n\), the last \(n\) digits alltogether must be divisible by \(2^n\). For powers of \(3\), the sum of digits of the dividend must be divisible by the divisor. Now, there's probably a divisibility rule for every prime number, but in some cases, it's better to just divide then spend time memorizing dozens of formulas. A good rule of thumb is to know up to \(10\). The only one we're missing right now is \(7\). This onea bit odd, but what you do is take the units digit, , double it, subtract it from the original number WITHOUT a units digit (so reducing each base), and keep doing that until you get a number you know is or isn't divisilbe by \(7\). Most importantly, for any non-prime integer, the divisibility rule is that it must satisfy the divisibility rules of all the prime factors.
Check an Integer is Divisible by \(7\)
Tratchenberg System For Multiplying By 12
There are a lot of quick multiplication, division, etc. tricks when it comes to math, if you want to go down that rabbit hole, but I want to focus on the Tratchenberg System. Tratchenberg was a mathematician known for creating a whole book and database essentially for quick mathematics, so you should check that out if it's something you're interested. One of his most memorable tricks was for multiplying by \(12\). This was one of the bigger ones that I learned, and thus one of the ones I wish to impart onto you.
Step One: Start with the units digit and multiply it by \(2\)
Step Two: If there was a tens digit, remember it and save it away for later. Now we look at the whole number's ten's digit. Multiply by \(2\) and add the \(1\) you carried before, if you did. Here's where it gets interesting: you "add the neighbor", or the value to the right
Step Three: Keep carying and adding the neighbor until you reach the end
What this does is essentially streamline the process of doing typical, vertical multiplication. Tratchenberg noticed that when you multiply by the tens digit, you add that value to the value to the left, so that's where "adding the neighbor comes from
Conclusion
Arithmetic is the fundamental for solving anything. Even if you know every trick in factoring and algebraic manipulation, to solve, you need to solve with arithmetic. Although the basic operations are fundamental and covered in elementrary school, quick computation is nescessary to be successful and save time on the AMC 10.
Question 1:
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