🔣 Number Properties
Odd & even, digit sums, and special number tricks
🔢 Odd and even
Even numbers end in 0, 2, 4, 6, or 8. Odd numbers end in 1, 3, 5, 7, or 9.
Understanding how odd and even combine is powerful for eliminating wrong answers:
- even + even = even
- odd + odd = even
- odd + even = odd
- even × anything = even
- odd × odd = odd
➕ Digit sums
The digit sum is what you get when you add all the digits of a number.
Example: The digit sum of 385 is 3 + 8 + 5 = 16. Since 16 is two digits, add them again: 1 + 6 = 7. So the digit sum of 385 is 7.
Digit sums reveal divisibility:
- Divisible by 3: A number is divisible by 3 if its digit sum is divisible by 3.
- Divisible by 9: A number is divisible by 9 if its digit sum is divisible by 9.
Why? When you divide by 10, the remainder is always 1 (because 10 ≡ 1 mod 3 and mod 9). So 385 = 3×100 + 8×10 + 5 has the same remainder as 3 + 8 + 5 when divided by 3 or 9.
🔄 Place value and digit swaps
Swapping digits changes a number completely. A two-digit number with tens digit T and units digit U equals 10T + U.
Example: The number 73 has T = 7 and U = 3, so its value is 10(7) + 3 = 73. When you swap to get 37, the value is 10(3) + 7 = 37.
The difference: 73 − 37 = (10T + U) − (10U + T) = 9T − 9U = 9(T − U) = 9(7 − 3) = 9(4) = 36.
⚡ Divisibility shortcuts
Quick tests to check divisibility without long division:
- Divisible by 2: Last digit is even (0, 2, 4, 6, 8)
- Divisible by 5: Last digit is 0 or 5
- Divisible by 10: Last digit is 0
- Divisible by 4: The last two digits form a number divisible by 4 (e.g., 316: check 16 ÷ 4 = 4 ✓)
- Divisible by 3: Digit sum is divisible by 3 (from above)
- Divisible by 9: Digit sum is divisible by 9 (from above)