Question

A number is always divisible by $90$ if
A) It is divisible by both $2$ and $45$
B) It is divisible by both $5$ and $18$
C) It is divisible by both $9$ and $10$
D) All of the above

Answer 1

Hint:If a number is divisible by $a$ and $b$, where $a$ and $b$ are co-primes, then that number will always be divisible by $ab$.Write all the possible values of two numbers which when multiplied to get 90 and from those two numbers choose the co-primes to get required answer.

Complete step-by-step answer:
According to the question, it is saying that a number is always divisible by $90$, then it must be divisible by which of the following numbers as given in the option.
So, here we know that if a number is divisible by $a$ and $b$, where $a$ and $b$ are co-primes, then that number will always be divisible by $ab$.
Now you might be wondering about what co-prime numbers are.
Basically you know about prime numbers that are those numbers which are divisible by one or itself. Similarly co-prime is called when the two numbers are given such that they do not have a common factor other than one, then that two numbers are called co-prime numbers.
Let us clear by taking some example
Like $9$ and $13$ are two numbers.
$9$ has factors $1,3,9$
$13$ has factors $1,13$ only.
So, the common factor is only one, therefore both are co-prime numbers.
If we take $2$ and $10$
$2$ has factors $1,2$
$10$ has factors $1,2,5,10$
So it has a common factor $1$ as well as $2$.
So, it is not a co-prime number.
Now according to question, we need to find total possible two numbers, which when multiplied get equal to $90$
$90 = 2 \times 45$ here, $2$ and $45$ are the numbers.
$90 = 3 \times 30$ here, $3$ and $30$ are the numbers.
$90 = 5 \times 18$ here, $5$ and $18$ are the numbers.
$90 = 6 \times 15$ here, $6$ and $15$ are the numbers.
$90 = 9 \times 10$ here, $9$ and $10$ are required numbers.
Now we had to determine the co-prime number among $\left( {2,45} \right),\left( {3,30} \right),\left( {5,18} \right),\left( {6,15} \right),\left( {9,10} \right)$
As we know co-prime numbers do not have common factors other than one.
Hence, $\left( {2,45} \right),\left( {5,18} \right),\left( {9,10} \right)$ are co-prime.
Hence we know that
If a number is divisible by $a$ and $b$, where $a$ and $b$ are co-primes, then that number will always be divisible by $ab$.
So, if a number is divisible by $90$, then it must be divisible by $2$ and $45$
It is also divisible by $5$ and $18$
And also divisible by $9$ and $10$
Because these three pairs are co-prime.
Hence, the answer is all of the above.

So, the correct answer is “Option D”.

Note:You must know the difference between prime numbers and co-prime numbers. And you should know that if a number is divisible by $a$ and $b$, where $a$ and $b$are co-primes, then that number will always be divisible by $ab$.