what is 21 divisible by

Guri Bee

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23-10-2024

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Is 21 Divisible by...? A Number Theory Exploration

The question "What is 21 divisible by?" might seem simple, but it delves into the fascinating world of number theory. Let's explore this question and uncover its hidden depths!

Divisibility Rules: The Fast Track

We can quickly determine if 21 is divisible by certain numbers using simple divisibility rules. These rules help us identify factors without performing long division:

• Divisible by 3: A number is divisible by 3 if the sum of its digits is divisible by 3. In the case of 21, 2 + 1 = 3, which is divisible by 3. So, 21 is divisible by 3.
• Divisible by 7: There's a handy trick for checking divisibility by 7: double the last digit and subtract it from the remaining digits. If the result is divisible by 7, the original number is also divisible by 7. For 21, double the last digit (1) to get 2, then subtract from 2 to get 0. Zero is divisible by 7, confirming that 21 is divisible by 7.

Finding All the Factors:

Using these divisibility rules, we know that 21 is divisible by 1, 3, 7, and 21 itself. However, are there any other factors?

Here's a systematic approach to finding all the factors of a number:

1. Start with 1 and the number itself: These are always factors.
2. Check divisibility by prime numbers: Prime numbers are only divisible by 1 and themselves. We've already checked for divisibility by 3 and 7.
3. Consider pairs: If a number is divisible by a prime number, it may also be divisible by a multiple of that prime number. For example, since 21 is divisible by 3, it could potentially be divisible by 6 (3 x 2). However, in this case, 21 is not divisible by 6.

Therefore, the complete set of factors of 21 are: 1, 3, 7, and 21.

Beyond the Basics:

The question of divisibility can be extended to explore deeper concepts:

• Prime Factorization: Every number can be expressed as a unique product of prime numbers. The prime factorization of 21 is 3 x 7.
• Greatest Common Factor (GCD): The GCD of two or more numbers is the largest number that divides into all of them. For example, the GCD of 21 and 35 is 7.
• Least Common Multiple (LCM): The LCM of two or more numbers is the smallest number that is a multiple of all of them. For example, the LCM of 21 and 35 is 105.

Conclusion:

By understanding divisibility rules and exploring further into number theory, we can gain a deeper appreciation for the properties of numbers. The seemingly simple question "What is 21 divisible by?" can lead to a fascinating journey of discovery!