eramsby1O1O

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We can analyze this problem by considering the divisibility rules for 2, 3, 4, 5, 6, 8, and 9:

 

Divisibility Rules:

 

A number is divisible by 2 if the last digit is even (0, 2, 4, 6, or 8).

 

A number is divisible by 3 if the sum of its digits is divisible by 3. (Given in the problem statement)

 

A number is divisible by 4 if the last two digits are divisible by 4. (Not applicable here since the number ends in 0)

 

A number is divisible by 5 if the last digit is a 0 or a 5.

 

A number is divisible by 6 if it's divisible by both 2 and 3.

 

A number is divisible by 8 if the last three digits are divisible by 8. (Not applicable here since the number ends in 0)

 

A number is divisible by 9 if the sum of its digits is divisible by 9. (This is similar to the divisibility rule for 3, but a stronger condition)

 

Analysis:

 

Ending in 0: Since the number ends in 0, it is automatically divisible by 2 and 5 (based on the divisibility rules).

 

Sum of Digits Divisible by 3: The problem states that the sum of the digits is divisible by 3. This means the number itself must also be divisible by 3 (based on the divisibility rule for 3).

 

Divisibility by Other Numbers:

 

Looking at the remaining numbers (4, 6, 8, and 9), we can see that:

 

4: Not necessarily divisible. Just because the last digit is 0 (divisible by 2), it doesn't guarantee divisibility by 4 (requires the last two digits to be divisible by 4).

 

6: Since the number is divisible by both 2 (ending in 0) and 3 (given in the problem statement), it is automatically divisible by 6.

 

8: Not necessarily divisible. Similar to 4, just because the number ends in 0 (divisible by 2), it doesn't guarantee divisibility by 8 (requires the last three digits to be divisible by 8).

 

9: Not necessarily divisible. The sum of digits being divisible by 3 is a weaker condition than being divisible by 9. So, a number divisible by 3 might not necessarily be divisible by 9.

 

Conclusion:

 

Therefore, a number ending in 0 with a sum of digits divisible by 3 is necessarily divisible by 2, 3, and 6 (out of the given numbers). There are 3​ such divisors.

May 25, 2024