Question11. True or false? Explain. a. If a counting number is divisible by 6 and 8 , it must be divisible by 48 . b. If a counting number is divisible by 4 , it must be divisible by 8 .
Studdy Solution
STEP 1
1. A counting number is a positive integer.
2. Divisibility by a number means that the counting number can be divided by that number without leaving a remainder.
3. We need to determine the truth value of each statement and provide an explanation.
STEP 2
1. Analyze statement (a).
2. Analyze statement (b).
STEP 3
Consider the first statement: "If a counting number is divisible by 6 and 8, it must be divisible by 48."
STEP 4
Find the least common multiple (LCM) of 6 and 8 to determine if divisibility by both implies divisibility by 48.
- The prime factorization of 6 is .
- The prime factorization of 8 is .
The LCM is found by taking the highest power of each prime that appears in the factorizations:
- The highest power of 2 is .
- The highest power of 3 is .
Thus, the LCM of 6 and 8 is .
STEP 5
Determine if divisibility by 24 implies divisibility by 48.
- A number divisible by 24 is not necessarily divisible by 48.
- For example, 24 is divisible by 6 and 8, but not by 48.
Therefore, the statement is false.
STEP 6
Consider the second statement: "If a counting number is divisible by 4, it must be divisible by 8."
STEP 7
Evaluate the divisibility condition.
- A number divisible by 4 is not necessarily divisible by 8.
- For example, 4 is divisible by 4 but not by 8.
Therefore, the statement is false.
The answers are:
a. False. A number divisible by 6 and 8 is not necessarily divisible by 48.
b. False. A number divisible by 4 is not necessarily divisible by 8.
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