Divisibility Rules: 5, 6, 8, 9, 10

Rule: a number is divisible by 5 if its last digit is 0 or 5.
- Expressed differently: the last digit ∈ {0, 5}.
Examples mentioned in the transcript:
- 20 ends in 0
- 1050 ends in 0
- 805 ends in 5
- 555 ends in 5
- -35 ends in 5
- -400 ends in 0
- 5 ends in 5
takeaway: ending in 0 or 5 guarantees that 5 is a factor.

Concept: 6 = 2 × 3. So a number must satisfy both the rule for 2 and the rule for 3 to be divisible by 6.
Rule for 2: an integer ends in 0, 2, 4, 6, or 8.
Rule for 3: the sum of the digits is divisible by 3.
Example from the transcript:
- 750 ends in 0 (passes the 2-rule).
- Sum of digits: 7 + 5 + 0 = 12, which is divisible by 3 (3 × 4).
- Since both tests pass, 750 is divisible by 6.
Note: If you pass both tests, you can conclude divisibility by 6; otherwise not.

The instructor notes that seven is conspicuously missing a simple, general rule in this discussion.
Recommendation: use a calculator if needed for checking divisibility by 7.
Rationale given: developing number sense and mental math helps performance on tests like the GRE.

Pattern for divisibility by powers of 2:
- For 2: look at the last digit.
- For 4: look at the last two digits.
- For 8: look at the last three digits.
Rule for 8: if the last three digits form a number divisible by 8, then the whole number is divisible by 8.
Example from the transcript:
- Number: 34,120
- Last three digits: 120
- Check: 120 is divisible by 8 since 120 = 8 \times 15.
Practical note: memorizing the 8 times table isn’t always necessary; you can use a calculator if you prefer.

Rule (analogous to 3): sum of the digits must be divisible by 9.
Process: sum the digits of the integer; if that sum is divisible by 9, then the original number is divisible by 9.
The transcript describes the process as: sum the digits, then check divisibility by 9 of that sum.
Example (illustrative): for the number 99, the digit sum is $9 + 9 = 18$, and 18 is divisible by 9, so 99 is divisible by 9.

Rule: a number is divisible by 10 if and only if it ends in 0.
Examples mentioned in the transcript:
- 300 ends in 0
- 200 ends in 0
- 20 ends in 0
- 450 ends in 0
Takeaway: ending with a 0 guarantees that 10 is a factor.
Quick note: this rule aligns with the general principle that a number is divisible by 10 precisely when its last digit is 0.

The video builds on earlier divisibility rules (1–4) and shows how to combine known rules (e.g., 6 = 2 × 3) to test other numbers.
Pattern recognition is highlighted for rules like 8 (look at the last three digits) to avoid memorizing large multiplication tables.
Practical advice given: use calculators when mental math slows you down, especially for larger numbers or less common divisibility checks.
Real-world relevance: mental math skills and quick divisibility checks help performance on standardized tests like the GRE and improve overall number sense.

Divisible by 5: last digit ∈ {0, 5}.
Divisible by 6: divisible by both 2 and 3. Tests: last digit even; sum of digits divisible by 3.
Divisible by 8: last three digits form a number divisible by 8.
Divisible by 9: sum of digits divisible by 9.
Divisible by 10: last digit = 0.

For 6 (example): 6 = 2 \times 3 and a number is divisible by 6 if it passes both the 2-rule and the 3-rule.
Example for 8 (from transcript): 120 = 8 \times 15 which confirms that a number ending with last three digits 120 is divisible by 8.
Example for 9 (conceptual): if the digit sum S = 7 + 2 + 9 = 18 and 18 is divisible by 9, then the original number is divisible by 9.
For 10 (conceptual): a number ending in 0 satisfies N \equiv 0 \pmod{10}, hence divisible by 10.