Is the number 34 divisible by 4? Can the first number be divided evenly by the second (without a remainder)? Compare the prime factorizations of the two numbers

Is the number 34 divisible by 4?

Method 1. The division of the two numbers:

A natural number 'A' could only be divisible by another number 'B' if after dividing 'A' by 'B' the remainder was zero.


34 would be divisible by 4 only if there was a natural number 'n', so that:
34 = 'n' × 4


When we divide the two numbers, there is a remainder:


34 ÷ 4 = 8 + remainder 2


There is no natural number 'n' such that: 34 = 'n' × 4.


The number 34 is not divisible by 4.


Note:

1) If you subtract the remainder of the above operation from the original number, 34, then the result is a number that is divisible by the second number, 4:


34 - 2 = 32


32 = 8 × 4


2) If you subtract the remainder of the above operation from the second number, 4, and then add the result to the original number, 34, you get a number that is divisible by the second number:

4 - 2 = 2


34 + 2 = 36.


36 = 9 × 4.


The number 34 is not divisible by 4
When the two numbers are divided, there is a remainder.

Method 2. The prime factorization of the numbers:

When are two numbers divisible?

The number 34 would be divisible by 4 only if its prime factorization contained all the prime factors that appear in the prime factorization of the number 4.


The prime factorization of the numbers:

The prime factorization of a number: finding the prime numbers that multiply together to make that number.


34 = 2 × 17
34 is not a prime number but a composite one.


4 = 22
4 is not a prime number but a composite one.



* The natural numbers that are only divisible by 1 and themselves are called prime numbers. A prime number has exactly two factors: 1 and itself.
* A composite number is a natural number that has at least one other factor than 1 and itself.


The prime factorization of the number 34 does not contain (all) the prime factors that occur in the prime factorization of 4.


The number 34 is not divisible by 4.


The number 34 is not divisible by 4.


The final answer:
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The number 34 is not divisible by 4.
When the two numbers are divided, there is a remainder.
The prime factorization of the number 34 does not contain (all) the prime factors that occur in the prime factorization of 4.

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The divisibility of the natural numbers:

Method 1: Divide the numbers and check the remainder of the operation. If the remainder is zero, then the numbers are divisible.

Method 2: The prime factorization of the numbers (the decomposition of the numbers into prime factors).

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