Number Theory Fundamentals
Divisibility rules are shortcuts that let you quickly check if one number is divisible by another without actually performing the whole long division.
Divisibility Rules
These as shortcuts that save you time.
Divisible by 2: A number is divisible by 2 if its last digit is even (0, 2, 4, 6, 8)
- Example: 4,582, ends in 2, divisible by 2
Divisible by 3: A number is divisible by 3 if the sum of its digits is divisible by 3
- Example: 4,582, 4 + 5 + 8 + 2 = 19, not divisible by 3, not divisible by 3
Divisible by 4: A number is divisible by 4 if its last two digits form a number divisible by 4
- Example: 4,582, last two digits: 82, 82 ÷ 4 = 20.5, not divisible by 4
Divisible by 5: A number is divisible by 5 if its last digit is 0 or 5
- Example: 4,582, ends in 2, not divisible by 5
Divisible by 6: A number is divisible by 6 if it's divisible by BOTH 2 and 3
- Example: 4,582, divisible by 2, but not by 3, so not divisible by 6
Divisible by 9: A number is divisible by 9 if the sum of its digits is divisible by 9
- Example: 4,582, 4 + 5 + 8 + 2 = 19, not divisible by 9, not divisible by 9
Divisible by 10: A number is divisible by 10 if its last digit is 0
- Example: 4,582, ends in 2, not divisible by 10
Divisible by 11: A number is divisible by 11 if the alternating sum of its digits is divisible by 11
- Alternating sum means: first digit - second digit + third digit - fourth digit, etc.
- Example: 4,582, 4 - 5 + 8 - 2 = 5, 5 is not divisible by 11, so not divisible by 11
Real Math Question
What is i^4582?
To solve this problem, we have to do two important things.
- Find a pattern of i(the imaginary number),
- Fit in 4582 into it.
Well, i has an important property, where i = √-1, i^2 = -1, i^3 = -√-1(or -i), and i^4 = 1.
The important thing to realize, is that this REPEATS. (so if we even had like i^7, it'd simply be the same thing has i^3)
So, if we can find out what number 4852 is divisible by, we can simply plug this into our above identity!
Well, it IS divisible by 1, and 2!(use the above rules to speed this process up)
Since 2 is bigger, we plug it into the identity, to derive that i^4852 is equal to:
-1
Why These Rules Matter:
- Quicker Math: Check divisibility instantly without a calculator
- Factorization: Break down numbers faster to find prime factors that work
- Problem Solving: Useful in competitive math, like shown in the problem above(I have gotten that very problem, and this rules have helped!)