- #1
Natasha1
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Without using a calculator, how can I know if 5 is a divisor of 3^444 and/or 4^333?
Is 5 a divisor of 3^444 & 4^333?
- Thread starter Natasha1
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In summary, the conversation discusses the use of Fermat's Little Theorem to determine if 5 is a divisor of 3^444 and/or 4^333 without using a calculator. It also mentions the concept of prime factorization and how any number can be written as a product of primes. The conversation ends with a discussion of the prime factorization of 3444 and 4333.
- #2
matt grime
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Fermat's Little theorem, how else can you say anything about anything?
- #3
Manchot
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Did you learn about prime factorization in grade school? 
- #4
matt grime
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true, i misread it.
- #5
Natasha1
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Manchot said:Did you learn about prime factorization in grade school?
No... :uhh:
- #6
HallsofIvy
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Well, you should have! Any number can be written, in a unique way, as a product of primes. How would 3444 be written as a product of primes? Any 5s in there? (Hint: 3 and 5 are prime numbers.)
How would 4333 be written as a product of primes? (Hint: 4 is not a prime number but 2 is.)
How would 4333 be written as a product of primes? (Hint: 4 is not a prime number but 2 is.)
1. Is 5 a divisor of 3^444?
No, 5 is not a divisor of 3^444. This is because 3^444 is an odd number and 5 is an even number, meaning there is no whole number that can divide evenly into both.
2. Is 5 a divisor of 4^333?
Yes, 5 is a divisor of 4^333. This is because 4^333 is an even number and 5 is also an even number, meaning there is a whole number (5) that can divide evenly into both.
3. How can you tell if a number is a divisor of a larger number?
To determine if a number is a divisor of a larger number, you can divide the larger number by the potential divisor. If the result is a whole number, then the potential divisor is indeed a divisor of the larger number. If the result is a decimal or fraction, the potential divisor is not a divisor of the larger number.
4. Why does the exponent of a number affect its divisibility?
The exponent of a number affects its divisibility because it determines the number's magnitude or size. For example, a larger exponent means a larger number and therefore more potential divisors. A smaller exponent means a smaller number and therefore fewer potential divisors.
5. Can a number be a divisor of both 3^444 and 4^333?
Yes, a number can be a divisor of both 3^444 and 4^333. For example, 1 is a divisor of both numbers because any number raised to the power of 0 is equal to 1. Additionally, 1 is a divisor of all numbers. Another example is 2, which is a divisor of both numbers because 2 is a common factor of 3 and 4.
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