The prime factorization of a number is 3 squared times 5 cubed times 7. Which statement is true about the factors of the number? Twenty-one is a factor of the number because both 3 and 7 are prime factors.
Twenty-one is not a factor of the number because 21 is not prime.
Ninety is a factor of the number because 3 squared = 9 and 90 is divisible by 9.
Ninety is not a factor of the number because 90 is not divisible by 7.

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.

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