Question

Is 49 a prime number? Can you prove it?

Short answer

No, 49 is not a prime number. It is divisible by 7.

Step-by-step solution

Understand the Definition of a Prime Number

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. Essentially, it should have exactly two distinct divisors.

Identify Divisibility Rules

To determine if 49 is a prime number, check if it can be divided by any integer other than 1 and 49 without a remainder.

Test Divisibility

Start with smaller prime numbers like 2, 3, 5, and 7. Since 49 is odd, it’s not divisible by 2. The sum of digits (4 + 9 = 13) is not divisible by 3, so 49 is not divisible by 3. 49 doesn’t end in 0 or 5, so it’s not divisible by 5. Finally, test divisibility by 7: Calculate: 49 ÷ 7 = 7 Therefore, 49 is divisible by 7.

Conclusion

Since 49 can be divided evenly by 7 (other than 1 and 49), it has more than two distinct positive divisors (1, 7, 49). Hence, 49 is not a prime number.

Key concepts

divisibility rules

Divisibility rules are shortcuts that help you determine if one number can be divided by another without doing long division. These rules save time and make problem-solving more efficient. Let's break down the rules applied to check if 49 is a prime number:
• **Rule for 2**: A number is divisible by 2 if it is even. Since 49 is odd, it fails this rule.
• **Rule for 3**: A number is divisible by 3 if the sum of its digits is divisible by 3. For 49, the sum is 4 + 9 = 13, which is not divisible by 3.
• **Rule for 5**: A number is divisible by 5 if it ends in a 0 or 5. Since 49 ends in 9, it fails this rule.
• **Rule for 7**: To check if 49 is divisible by 7, we divide 49 by 7 and get 7, which is a whole number. Hence, 49 is divisible by 7.
When you apply these basic divisibility rules, you quickly see that 49 is not a prime number because it is divisible by 7.

prime number definition

Prime numbers have fascinated mathematicians for centuries. A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. This means a prime number must have exactly two distinct positive divisors. Here are some examples:
• **Prime Numbers**: 2, 3, 5, 7, 11, 13, 17
• **Non-Prime Numbers**: 4 (divisible by 1, 2, 4), 6 (divisible by 1, 2, 3, 6), 9 (divisible by 1, 3, 9), and in our example, 49 (divisible by 1, 7, 49).
To determine if a number is prime, you must check if it is only divisible by 1 and itself. This checking process often involves testing divisibility by smaller prime numbers, as shown in the exercise.

mathematics problem-solving

Mathematics problem-solving is a crucial skill that allows you to tackle complex questions systematically. The steps used in determining if 49 is a prime number offer a great example of this method:
1. **Understand the Problem**: Begin by understanding what a prime number is.
2. **Gather Information**: Use divisibility rules to test if the number has any divisors besides 1 and itself.
3. **Test and Apply Knowledge**: Practically check each rule, starting from smaller prime numbers.
4. **Conclude the Solution**: Based on the divisors found, determine whether the number is prime or not.
This structured approach not only helps in solving mathematical problems but also fosters critical thinking and logical reasoning skills essential in various aspects of life.