Counting Positive Factors

12 has 6 positive divisors: 1, 2, 3, 4, 6, 12.

Which of the two numbers has more divisors, 12 × 6 12\times 6 or 12 × 9 ? 12\times 9?

12 × 6 12 \times 6 12 × 9 12 \times 9 Both have an equal number of divisors.

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2 solutions

Sathvik Acharya
Jun 3, 2017

Let us first prime factorize the numbers that we are going to compare.

12 × 6 = 2 3 × 3 2 12\times 6=2^3\times 3^2 12 × 9 = 2 2 × 3 3 12\times 9=2^2\times 3^3

So, the number of factors in 12 × 6 = 2 3 × 3 2 12\times 6=2^3\times 3^2 is ( 3 + 1 ) ( 2 + 1 ) = 4 × 3 = 12 (3+1)(2+1)=4\times 3=\boxed{12} and 12 × 9 = 2 2 × 3 3 12\times 9=2^2\times 3^3 is ( 2 + 1 ) ( 3 + 1 ) = 3 × 4 = 12 (2+1)(3+1)=3\times 4=\boxed{12} .

Therefore, both 12 × 6 12\times 6 and 12 × 9 12\times 9 have the equal number of divisors .

To learn how to calculate the number of divisors of a number click here

I legit still didn't get it

Ethan Weng - 9 months, 2 weeks ago

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I would suggest learning how to calculate the number of divisors of number. The solution might make more sense after that.

Sathvik Acharya - 9 months, 2 weeks ago
Mark Geha
Jun 10, 2017

Just by writing out the divisors, 108 has 1x108, 2x54, 3x36, 4x27, 6x18 and 9x12 - 12 in total. 72 has 1x72, 2x36, 3x24, 4x27, 6x18 and 9x12 - also 12 in total.

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