To find the factors of 363, we need to determine all the positive integers that divide 363 without leaving a remainder. Let's go through the process step-by-step.
1. Start with the smallest possible factor, which is 1:
- [tex]\( 363 \div 1 = 363 \)[/tex]
- So, 1 is a factor of 363.
2. Next, check if 363 is divisible by 2:
- 363 is an odd number, and therefore, it cannot be divided evenly by 2.
- So, 2 is not a factor of 363.
3. Check if 363 is divisible by 3:
- To do this quickly, sum the digits of 363: [tex]\( 3 + 6 + 3 = 12 \)[/tex]
- Since 12 is divisible by 3, 363 is also divisible by 3.
- [tex]\( 363 \div 3 = 121 \)[/tex]
- So, 3 is a factor of 363, and its quotient, 121, is also a factor.
4. Check if 363 is divisible by 4:
- 363 is not an even number, so it cannot be divided evenly by 4.
- So, 4 is not a factor of 363.
5. Next, check 5, 6, 7, 8, 9, and 10:
- 363 does not end in 0 or 5, so it's not divisible by 5.
- The sum of the digits (12) is not divisible by 3 twice over, so it's not divisible by 6.
- There are no quick tricks for 7, 8, 9, and 10, but we can test division and determine that 363 is not divisible evenly by any of these numbers.
6. Check if 363 is divisible by 11:
- There is a trick for divisibility by 11: Taking the alternative sum of the digits: [tex]\( 3 - 6 + 3 = 0 \)[/tex]
- Because 0 is divisible by 11 (i.e., 0 mod 11 = 0), 363 is divisible by 11.
- [tex]\( 363 \div 11 = 33 \)[/tex]
- So, 11 is a factor of 363, and its quotient, 33, is also a factor.
7. Finally, check if 363 is divisible by larger factors up to the square root of 363 (approximately 19):
- We don't need to test the multiples of factors we've already checked.
- Continuing from above, detailed testing reveals that no additional factors exist beyond those already determined.
8. List all the discovered factors:
- The factors of 363 are the numbers that divide it evenly: 1, 3, 11, 33, 121, and 363.
Thus, the complete list of factors of 363 is:
[tex]\[ 1, 3, 11, 33, 121, 363 \][/tex]
