
Find the number of odd proper divisors of \[{{3}^{p}}{{6}^{m}}{{21}^{n}}\].
A. $\left( P+1 \right)\left( m+1 \right)\left( n+1 \right)-2$
B. $\left( P+m+n+1 \right)\left( n+1 \right)-1$
C. $\left( P+1 \right)\left( m+1 \right)\left( n+1 \right)-1$
D. None of these
Answer
505.5k+ views
Hint: If any problem where we have to find proper divisors, we first Express that number into its prime factors i.e. we prime factorize the number so here also we prime factorize the number \[{{3}^{p}}\times {{6}^{m}}\times {{21}^{n}}\]. After getting the prime factors we reject even factors as we only want odd proper divisors.
The proper divisor is a divisor of a number excluding the number itself.
Complete step by step solution: Let \[N={{3}^{p}}\times {{6}^{m}}\times {{21}^{n}}\]
first we do the prime factorization of N:
\[\therefore N={{3}^{p}}\times {{\left( 2\times 3 \right)}^{m}}\times {{\left( 3\times 7 \right)}^{n}}\]
\[={{3}^{p}}\times {{2}^{m}}\times {{3}^{m}}\times {{3}^{n}}\times {{7}^{n}}\]
\[={{2}^{m}}\times {{3}^{p+m+n}}\times {{7}^{n}}\]
\[\therefore N={{2}^{m}}\times {{3}^{p+m+n}}\times {{7}^{n}}\]
Since we want only odd proper divisors ,so we don’t want 2 as one of its divisors. Hence, we will reject ${2}^{\text{m}}$ as the proper divisor.
We have:
$\text{N}=2^0 \times {3}^{\text{m+p+n}}\times 7^{\text{n}}$
which simplifies to,
$\text{N}=1 \times 3^{\text{a}} \times 7^{\text{b}}$, where: a = m + p + n and, b = n
Now, since ‘a’ can be any integer between 0 to p+m+n and ‘b’ can be any integer from 0 to n, we have 0≤a≤p+m+n and 0≤b≤n.
This implies ‘a’ can take total p+m+n+1 values and ‘b’ can take n+1 values. But, we cannot include the given number as one of its proper divisors.
Therefore,
Total number of odd proper divisors=(p+m+n+1)(n+1)-1
Therefore, Correct option is B.
Note: There is one point that you must remember as you can go wrong there that we are rejecting ${2}^{\text{m}}$ as one of its proper divisors because we only want the odd proper divisors. If we include ${2}^{\text{m}}$ then, even proper divisors will also get included in the calculation. So, this is where you must be very careful about this problem.
The proper divisor is a divisor of a number excluding the number itself.
Complete step by step solution: Let \[N={{3}^{p}}\times {{6}^{m}}\times {{21}^{n}}\]
first we do the prime factorization of N:
\[\therefore N={{3}^{p}}\times {{\left( 2\times 3 \right)}^{m}}\times {{\left( 3\times 7 \right)}^{n}}\]
\[={{3}^{p}}\times {{2}^{m}}\times {{3}^{m}}\times {{3}^{n}}\times {{7}^{n}}\]
\[={{2}^{m}}\times {{3}^{p+m+n}}\times {{7}^{n}}\]
\[\therefore N={{2}^{m}}\times {{3}^{p+m+n}}\times {{7}^{n}}\]
Since we want only odd proper divisors ,so we don’t want 2 as one of its divisors. Hence, we will reject ${2}^{\text{m}}$ as the proper divisor.
We have:
$\text{N}=2^0 \times {3}^{\text{m+p+n}}\times 7^{\text{n}}$
which simplifies to,
$\text{N}=1 \times 3^{\text{a}} \times 7^{\text{b}}$, where: a = m + p + n and, b = n
Now, since ‘a’ can be any integer between 0 to p+m+n and ‘b’ can be any integer from 0 to n, we have 0≤a≤p+m+n and 0≤b≤n.
This implies ‘a’ can take total p+m+n+1 values and ‘b’ can take n+1 values. But, we cannot include the given number as one of its proper divisors.
Therefore,
Total number of odd proper divisors=(p+m+n+1)(n+1)-1
Therefore, Correct option is B.
Note: There is one point that you must remember as you can go wrong there that we are rejecting ${2}^{\text{m}}$ as one of its proper divisors because we only want the odd proper divisors. If we include ${2}^{\text{m}}$ then, even proper divisors will also get included in the calculation. So, this is where you must be very careful about this problem.
Recently Updated Pages
Master Class 9 General Knowledge: Engaging Questions & Answers for Success

Earth rotates from West to east ATrue BFalse class 6 social science CBSE

The easternmost longitude of India is A 97circ 25E class 6 social science CBSE

Write the given sentence in the passive voice Ann cant class 6 CBSE

Convert 1 foot into meters A030 meter B03048 meter-class-6-maths-CBSE

What is the LCM of 30 and 40 class 6 maths CBSE

Trending doubts
The largest brackish water lake in India is A Wular class 9 biology CBSE

Define human made resources

What is the role of Mahatma Gandhi in national movement

Which Army is not a professional occupation A Indian class 9 social science CBSE

The central location of India at the head of the Indian class 9 social science CBSE

What subjects did Margie and Tommy learn class 9 english CBSE
