Shortcut Trick
Total natural numbers from 1 to 19 = 19
Prime numbers from 1 to 19 are 2, 3, 5, 7, 11, 13, 17, 19 (Total = 8)
The number 1 is a unique number (neither prime nor composite).
Composite numbers = Total numbers − Prime numbers − 1
⇒ Composite numbers = 19 − 8 − 1 = 10
∴ The correct answer is 10.
Alternate Method
Given:
The range of natural numbers starting from 1 up to 19.
Concept Used:
A composite number is any positive integer that can be formed by multiplying two smaller positive integers. It must have at least one divisor other than 1 and itself.
The number 1 is an exception as it is classified as neither prime nor composite.
Calculation:
⇒ Let us list out all the natural numbers from 1 to 19:
⇒ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19
⇒ Identify and eliminate 1, because it is not a composite number.
⇒ Identify and eliminate all the prime numbers (numbers with exactly two distinct positive divisors):
⇒ Primes to remove: 2, 3, 5, 7, 11, 13, 17, 19
⇒ The remaining numbers in our sequence are all composite:
⇒ 4, 6, 8, 9, 10, 12, 14, 15, 16, 18
⇒ Counting this final set of numbers gives exactly 10 composite numbers.
∴ The correct answer is 10.
Additional Information
Prime Numbers
A natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers. Example: 2, 3, 5, 7, 11.
Composite Numbers
A positive integer that has at least one positive divisor other than 1 or itself. Example: 4, 6, 8, 9.
Property of 1
The number 1 has only one positive divisor (itself), making it uniquely classified as neither prime nor composite.
Smallest Composite Numbers
The smallest even composite number is 4, and the smallest odd composite number is 9.