Answer:
Twenty-one is a factor of the number because both 3 and 7 are prime factors.
Step-by-step explanation:
Given number is,
3^2\times 5^3\times 73
2
×5
3
×7
=3\times 3\times 5\times 5\times 5\times 7=3×3×5×5×5×7
Where, 3, 5 and 7 are prime numbers ( only divisible by 1 and itself ),
⇒ Both 3 and 7 are prime factors of the given number,
⇒ 21 is a factor of the given number.
Thus, first option is correct.
⇒ Second option is incorrect.
Now, 5 is factor of the given number but 2 is not,
⇒ 10 is not a factor of the given number,
⇒ 90 is not a factor of the given number,
⇒ Third option is incorrect.
Suppose 90 is divisible by 7,
⇒ 90 = 7a
Where a is any whole number,
⇒ 7=\frac{90}{a}7=
a
90
3^2\times 5^3\times 7=3^2\times 5^3\times \frac{90}{a}3
2
×5
3
×7=3
2
×5
3
×
a
90
Since, 90 could be a factor of this number, if a = 3 or 5 or their multiple,
For the other values of a, 90 can not be the factor,
Hence, there is no effect of divisibility of 90 by 7 on having 90 as a factor of the given number,
⇒ Fourth option is incorrect.
Considering the prime factorization of the question described above, the statement that is true is "Twenty-one is a factor of the number because both 3 and 7 are prime factors."
The evidence is shown below:
3^2 multiply by 5^3 multiply by 7
=> 3 multiply by 3 multiply by 5 multiply by 5 multiply by 5 multiply by 7
=> 3×3×5×5×5×7
Here, we can see that 3, 5, and 7 are prime numbers, which implies that they are only divisible by one and themselves.
Since both 3 and 7 are prime factors of the given number.
Therefore, 21 is a factor of the given number.
Hence, in this case, it is concluded that the correct answer is option A. "Twenty-one is a factor of the number because both 3 and 7 are prime factors."
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