How can we find out if a number is divisible by 6? Will checking its divisibility by its factors 2 and 3 work? Use the shortcuts for 2 and 3 on these numbers and divide each number by 6 to verify—38, 225, 186, 64.
Yes, a number is divisible by 6 if and only if it is divisible by both 2 and 3.
38: Divisible by 2 (ends in 8) Not divisible by 3 (sum = 11) ∴ Not divisible by 6
225: Not divisible by 2 (ends in 5) ∴ Not divisible by 6
186: Divisible by 2 (ends in 6) Divisible by 3 (sum = 15) Verification: 186 ÷ 6 = 31 ∴ Divisible by 6
64: Divisible by 2 (ends in 4) Not divisible by 3 (sum = 10) ∴ Not divisible by 6
Explanation
The chapter explains that since 6 = and 2 and 3 are coprime (no common factors), a number is divisible by 6 if and only if it is divisible by both 2 and 3. This works because divisibility by 2 and 3 together ensures the prime factorization contains both 2 and 3, which means it contains 6. Verification: 38÷6=6.33 (not divisible), 225÷6=37.5 (not divisible), 186÷6=31 (divisible), 64÷6=10.67 (not divisible).
Solution Steps
Check if the number is divisible by 2 (the number must end in 0, 2, 4, 6, or 8).
Check if the number is divisible by 3 (the sum of the digits must be divisible by 3).
Apply the rule: If the number is divisible by both 2 and 3, then it is divisible by 6.
Verify the result by dividing the number by 6.