Question

The sum of non-prime positive divisors of 450 is,
A.$1209$
B.$1299$
C.$1199$
D.$1109$

Answer 1

Hint: We have to find the sum of non-prime positive divisors of 450. We will find the prime factors and make the combinations in order to get the answer.

Complete step-by-step answer:
A prime number is a number which has exactly two factors i.e. it can be divided by only the number ‘1’ and itself. A composite number has more than two factors, which means apart from getting divided by number 1 and itself, it can also be divided by at least one integer or number. We don’t consider the number ‘1’ as a composite number. Also the number “1” is not a prime number.
A prime number is a positive integer having exactly two factors. If p is a prime number, then it’s only factors are necessarily 1 and p itself. Any number which does not follow this is termed as a composite number which means that they can be factored into other positive integers.
To understand the concept of prime and composite numbers, let us see the following examples. Factors of 6 are 1, 2, 3 and 6, which are four factors in total. But factors of 7 are only 1 and 7 totally two. Hence, 7 is a prime number but 6 is not, instead it is a composite number. But always remember that 1 is neither prime nor composite.
The integers which can be generated by multiplying the two smallest positive integers and contains at least one divisor other than number ‘1’ and it is known as composite numerals or numbers. It is neither a prime number nor a unit. This is the general difference between composite and prime numbers.
Prime factorization of 450, are $2\times 3\times 3\times 5\times 5$, now non-prime divisors will be combinations of these. When two of these are multiplied, $2\times 3=6,2\times 5=10,3\times 3=9,3\times 5=15,5\times 5=25$.
When three of these are multiplied, $2\times 3\times 3=18,2\times 3\times 5=30,2\times 5\times 5=50,3\times 3\times 5=45,3\times 5\times 5=75$.

When four of these are multiplied, $2\times 3\times 3\times 5=90,2\times 3\times 5\times 5=150,3\times 3\times 5\times 5=225$.
When four of these are multiplied, $2\times 3\times 3\times 5\times 5\times =450$.
Also 1 is also a non-prime divisor.
Sum of divisors $=1998+1=1199$
Therefore, the sum of non-prime positive divisors of 450 is 1199.
Therefore the correct answer is option (C).

Note: Read the question and see what is asked. Concept regarding prime numbers and composite numbers should be clear. A proper assumption should be made. Do not make silly mistakes while substituting. Equate it in a proper manner and don't confuse yourself.