Question

How many prime numbers between 1 and 100 are the factors of 7,150?

Answer 1

Hint: The given problem is related to prime factorization. Express the number as a product of prime factors. The number of prime factors between 1 and 100 is our required answer.

Complete step-by-step answer:
Before proceeding with the solution, let’s understand the concept of prime factorization. A prime number is a number which is not divisible by any other number except 1 and itself. Any number can be expressed as a product of prime numbers. All the prime numbers, which when multiplied, give a product equal to a number (say x) are called the prime factors of the number x. To write the prime factors of a number, we should always start with the smallest prime number, i.e. 2 and check divisibility. If the number is divisible by the prime number, then we write the number as a product of the prime number and another number, which will be the quotient when the given number is divided by the prime number. Then, we take the quotient and repeat the same process. This process is repeated till we are left with 1 as the quotient.
For example: Consider the number 51. It is an even number. So, it is not divisible by 2. The sum of the digits of 51 is 5 + 1 = 6. Hence, 51 is divisible by 3. Now, $51=3\times 17$ . Now, we take 17. We know, 17 is a prime number. Hence, the prime factors of 51 are 3 and 17.
Now, coming to the question, we are asked to find the prime factors of 7150, which lie between 1 and 100.
Now, 7150 is an even number. So, we can write 7150 as $7150=2\times 3575$ . Now, 3575 is an odd number and the sum of its digit is equal to 3 + 5 + 7 + 5 = 20, which is not divisible by 3. Hence , 3575 is not divisible by 2 or 3. Now, the unit digit of 3575 is 5 . So, it is divisible by 5. So, we can write 3575 as $3575=5\times 715$ . Again, we can write 715 as $715=5\times 143$ . Again, 143 can be written as $143=11\times 13$ . Now, 11 and 13 both are prime numbers. So, we can write 7150 as $7150=2\times 5\times 5\times 11\times 13$ . So, the number of prime factors of 7150, which lie between 1 and 100, is 4.

Note: While expressing the numbers as a product of prime factors, always start with the least prime number. It will be easier to find the prime factors.