Lisa had to find out the number of divisors of the natural number $46\,200$.
She solved the problem in the following steps:
(1) She made prime factor decomposition.
$$46\,200=2\cdot2\cdot2\cdot3\cdot5\cdot5\cdot7\cdot11$$
(2) She expressed the result using exponents.
$$46\,200=2^3\cdot3^1\cdot5^2\cdot7^1\cdot11^1$$
(3) She claimed that the number of its divisors is the sum of exponents obtained after a prime factor decomposition, i.e., $8$. $$3+1+2+1+1=8$$
Is her solution correct? If not, identify the incorrect step.
Yes. The whole solution is perfectly fine.
No, her solution is not correct. The mistake is in step (1). The prime factor decomposition is:
$$46\, 200= 3^1\cdot7^1\cdot8^1\cdot11^1\cdot25^1$$ The number of its divisors is: $$1+1+1+1+1=5$$
No, her solution is not correct. The mistake is in step (3). The total number of factors is found by adding 1 to each exponent and then summing these together. Therefore, the number of divisors is:
$$4+2+3+2+2=13$$
No, her solution is not correct. The mistake is in step (3). The total number of factors is found by adding $1$ to each exponent and then multiplying these together. Therefore, the number of divisors is: $$4\cdot2\cdot3\cdot2\cdot2=96$$
For a number whose prime factorization is $x^a\cdot y^b$, we determine the total number of factors by adding $1$ to each exponent and then multiplying these together. This expresses the number of factors formula as $(a + 1)\cdot(b + 1)$, where $a$ and $b$ are the exponents obtained after the prime factorization of the given number.
For example:
The number $12$ has a prime factor decomposition $12 =2^2\cdot3^1$.
The number of its divisors is $(2 + 1)\cdot(1 + 1) = 6$.
All divisors have the form $2^a\cdot3^b$, where $0 \leq a \leq 2$, $0 \leq b \leq 1$.
They are $1 = 2^0\cdot3^0$, $2 = 2^1\cdot3^0$, $3 =2^0\cdot3^1$, $4 = 2^2\cdot3^0$, $6 = 2^1\cdot3^1$,and $12 = 2^2\cdot3^1$.
Correct solution:
The number $46\, 200$ has a prime factor decomposition $46\, 200=2^3\cdot3^1\cdot5^2\cdot7^1\cdot11^1$.
The number of its divisors is (3 + 1) ∙ (1 + 1) ∙ (2 + 1) ∙ (1 + 1) ∙ (1 + 1) = 96.