Question

Check the validity of the following statement:
P: 60 is a multiple of 3 and 5.
A. true
B. false

Answer 1

Hint: We try to find out the concept of multiplier. we express the given number termed as a multiplication form of two other positive whole numbers. We also check whether it is directly divisible by 3 or 5 using the division formulas of 3 and 5.

Complete step-by-step solution:
The concept of a number is a multiple of another number comes from the concept of divisibility or expression of the first number being termed as a multiplication form of two other positive whole numbers different from the given first number.
So, we need to check the divisibility of 60 by 3 and 5.
For the divisibility of 3, we know that if the sum of the digits in a number is divisible by 3 then the number itself is divisible by 3.
For example, we take a number (abc) where a, b, c are the digits in that number in the hundredth, tenth, unit places. So, we find $a+b+c$. If the sum is divisible by 3 then (abc) is divisible by 3. Take 4747. We add up the digits and get $4+7+4+7=22$ which is not divisible by 3. So, 4737 is not divisible by 3 where $\dfrac{4747}{3}=1582.3 $.
For the divisibility of 5, we know that if the last digit i.e. the unit place digit is 0 or 5 then the number itself is divisible by 5.
For example, we take a number (abc) where a, b, c are the digits in that number in the hundredth, tenth, unit places. So, we find c. If c is 0 or 5 then (abc) is divisible by 5. Take 4735. We have 5 in unit place, so it’s divisible by 5. So, 4735 is divisible by 5 where $\dfrac{4735}{5}=947$.
Now we talk about 60. The sum of the digits is $6+0=6$ which is divisible by 3. So, 60 is divisible by 3.
60 has 0 in its unit place. So, 60 is also divisible by 5.
Now we express 60 as a multiplication of two positive whole numbers.
So, $60=3\times 20$. In this factorization of 60, we have two positive whole numbers and 3 and 20 which means 60 is a multiple of 3 when it’s multiplied by 20.
Again $60=5\times 12$. In this factorization of 60, we have two positive whole numbers and 5 and 12 which means 60 is a multiple of 5 when it’s multiplied by 12.
So, the given statement is true. The correct option is (A).

Note: We need to remember the theorems for the divisibility. If we can’t then there will be no way to find out other than doing long division. Using the theorem saves a lot of time. The multiplication form is also very important to establish the statement as true.