# What are some simple shortcuts to finding the number of factors that a number has?

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Not what all the factors are, but how *many* factors there are to that number.

And, *please* make it simple but short and understandable.

Not what all the factors are, but how *many* factors there are to that number.

And, *please* make it simple but short and understandable.

See video.

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There are no shortcuts, but there are ways by which one can identify, whether a given number is divisible by a prime number or not, which helps in factorizing a number. However, this is only for smaller numbers, say up to

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One shortcut to find the number of factors a given number has is to first express the number as a product of its prime factors. Then, add 1 to each exponent in the prime factorization and multiply these incremented exponents together. This will give you the total number of factors.

Another shortcut is to use the divisor function. This function, denoted as ( d(n) ), gives the number of positive divisors of ( n ). It can be computed by finding the prime factorization of ( n ), then adding 1 to each exponent and multiplying these incremented exponents together.

For example, let's say you have the number 36:

- Express 36 as a product of prime factors: ( 36 = 2^2 \times 3^2 ).
- Add 1 to each exponent: ( (2+1) \times (2+1) = 3 \times 3 = 9 ).
- So, 36 has 9 factors.

Alternatively, you can use the divisor function directly:

[ d(n) = (a_1 + 1)(a_2 + 1)\ldots(a_k + 1) ]

where ( n = p_1^{a_1} \times p_2^{a_2} \times \ldots \times p_k^{a_k} ) is the prime factorization of ( n ).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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