Question

Find the value of p and q, so that the prime factorisation of 2520 can be expressed as ${2^3} \times {3^p} \times q \times 7$.

Answer 1

Hint: When a number is written in the form of the product of its all prime numbers e.g. 35 is written as a product of its prime factors i.e. 7 and 5. Hence, in prime factorisation it is represented as $1 \times 5 \times 7$.
To find the values of p and q, write 2520 in the prime factorisation form and then compare it with ${2^3} \times {3^p} \times q \times 7$.

Complete step-by-step answer:
First of all, we have given 2520 is expressed in prime factorisation form as ${2^3} \times {3^p} \times q \times 7$.
We have to find the value of p and q.
As in prime factorisation the number is written in the form of the product of its all prime factors. Therefore, we will find all the prime factors of number 2520 as follows:

2	2520
2	1260
2	630
3	315
3	105
5	35
7	7
	1

Here are the steps to explain the above table.
Again, it can be divided by 2 then we get 630.
Again, we divide it by 2 and we get 315.
Now, it can’t be divided by 2 as the last digit is odd i.e. 5 and if the last digit is odd then the number will not be divided by 2.
Hence, check for 3. For 3 the divisibility test says sum must be divided by 3. As the sum of digits in 315 is 9 which is divisible by 3. So, by dividing 315 with 3, we get 105.
Again, it is divisible by 3 as the sum of the digits is 6. So, by dividing 105 by 3, we get 35.
Now 35 is the multiple of 5 and 7.

Therefore, in prime factorisation form 2520 can be expressed as
$ \Rightarrow 2520 = 2 \times 2 \times 2 \times 3 \times 3 \times 5 \times 7$
Now, write this in the power form and we get,
$ \Rightarrow 2520 = {2^3} \times {3^2} \times 5 \times 7$
By comparing the above with ${2^3} \times {3^p} \times q \times 7$ we can see the power of 3 is p in ${2^3} \times {3^p} \times q \times 7$and in the above it is 2. So, $p = 2$and 5 is missing in ${2^3} \times {3^p} \times q \times 7$. Hence the value of q is 5.
Therefore, the value of $p = 2$ & $q = 5$.
Note: Let us suppose if 5 is also there in ${2^3} \times {3^p} \times q \times 7$ i.e. if they have given that 2320 in prime factorisation is represented as ${2^3} \times {3^p} \times q \times 5 \times 7$. Then in this case what will be the value of q? some of the students will answer 0 which is wrong. The correct answer is 1 as 1 is the prime factor of all numbers. To check your understanding this question can be asked many times.