Question11. True or false? Explain. a. If a counting number is divisible by 9 , it must be divisible by 3 . b. If a counting number is divisible by 3 and 11 , it must be divisible by 33 .
Studdy Solution
STEP 1
1. A counting number is a positive integer.
2. Divisibility rules for integers are applicable.
3. We need to determine the truth of each statement and provide an explanation.
STEP 2
1. Analyze statement (a).
2. Analyze statement (b).
STEP 3
Consider the divisibility rule for 9. A number is divisible by 9 if the sum of its digits is divisible by 9.
STEP 4
Consider the divisibility rule for 3. A number is divisible by 3 if the sum of its digits is divisible by 3.
STEP 5
Since a number divisible by 9 has a digit sum divisible by 9, and since 9 is a multiple of 3, the digit sum is also divisible by 3. Therefore, any number divisible by 9 is also divisible by 3.
STEP 6
Conclusion for statement (a): True. If a counting number is divisible by 9, it must be divisible by 3.
STEP 7
Consider the divisibility rule for 3 and 11. If a number is divisible by both 3 and 11, it means it has both 3 and 11 as factors.
STEP 8
The least common multiple (LCM) of 3 and 11 is 33. Therefore, if a number is divisible by both 3 and 11, it must be divisible by their LCM, which is 33.
STEP 9
Conclusion for statement (b): True. If a counting number is divisible by 3 and 11, it must be divisible by 33.
The answers are:
a. True
b. True
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