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## Comment on

Counting the Divisors of Large Numbers## great tip! thank you so much!

## Should we consider 1 as the

## 1 is a divisor of all

1 is a divisor of all positive integers. However, with respect to the technique described in this video, we wouldn't include 1 in the prime factorization of the number for which we're trying to find the total number of divisors.

For example, to determine the total number of divisors of 14000, we first prime factorize 14000 to get 14000 = (2^4)(5^3)(7^1)

We wouldn't write: 14000 = (1^1)(2^4)(5^3)(7^1)

We wouldn't do this because 1 is NOT a prime number, so it has no place in the PRIME factorization of a number.

Also, including 1 would be problematic, because the exponent could be ANY number.

For example, 12 = (1^1)(2^2)(3^1) or 12 = (1^4)(2^2)(3^1) or 12 = (1^33)(2^2)(3^1) etc.

Does that help?

## What is the rationale behind

## I cover this a 1:50 in the

I cover this a 1:50 in the video.

We add 1, because we must also consider the possibility that the exponent is 0.

So, for example, if 2^5 is in the prime factorization of N, then when it comes to possible divisors of N, we must consider 2^5, 2^4, 2^3, 2^2, 2^1 and 2^0.

So there are 5+1 possible exponents.

## I got 30 positive divisors

I got 20 for 3375 and 36 for 100 000. Where am I going wrong? Hmm.

## 3375 = (3)(3)(3)(5)(5)(5) =

3375 = (3)(3)(3)(5)(5)(5) = (3^3)(5^3)

So, the number of divisors = (3+1)(3+1) = 16

1,000,000 = (2)(2)(2)(2)(2)(2)(5)(5)(5)(5)(5)(5) = (2^6)(5^6)

So, the number of divisors = (6+1)(6+1) = 49

Does that help?

Cheers,

Brent

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