Method 1: decompose the original formula into prime factors, that is, write it as the product of prime factors. If you want to form 0 (or 10), you must look at the logarithm of 2 and 5 in this prime factor product formula, because a pair forms a zero.

It can be seen intuitively that the number of 2 is obviously more than 5, so just look at the number of 5. The number that can be decomposed into 5 in the formula is:

5、 10、 15、20、25、30、35、40、45、50、55、60、65、70、75、80、85、90、95、 100

The number of 5 is:

1、 1、 1、 1、 2、 1、 1、 1、 1、 2、 1、 1、 1、 1、 2、 1

1、 1、 1、 2

There are 24 * * * in total, so the total * * * will form 24 zeros.

Method 2: Multiply 1 0 consecutive integers from 10 to 10:

1×2×3×4×5×6×7×8×9× 10。

How many zeros are there at the end of a continuous product?

The answer is two zeros. Where, the factor 10 gives 1 zeros, and the factors 2 and 5 are multiplied to give 1 zeros, and two zeros count as * * *.

Just two zeros? Will there be more?

If you don't believe me, you can calculate the product and get the result.

Original formula =3628800. You see, there are exactly two zeros at the end of the product, not more than 65438+ zeros.

So, what if we expand the scale and lengthen the team? For example, from 1 times 20:

1×2×3×4×…× 19×20。 How many zeros are there at the end of the product?

Now the answer has become four zeros. Where we get 1 zeros from the factor 10, 1 zeros from 20, 1 zeros from 5 and 2, and 1 zeros from 15 and 4.

Just four zeros? Will there be more?

Don't worry, that's all. In order to get 0 at the end of the product, prime factor 5 and prime factor 2 must be multiplied in pairs. Among the prime factors of products, 2 is more and 5 is less. There is a prime factor of 5 and a 0 at the end of the product. From 1 multiplied by 20, only 5, 10, 15 and 20 each have a prime factor of 5, and there may be only four zeros at the end of the product, and there will be no more.

Expand the ratio a little, from 1 to 30:

1×2×3×4×…×29×30。 How many zeros are there at the end of the product now?

Obviously, there are at least six zeros.

You see, from 1 to 30, where 5, 10, 15, 20, 25 and 30 are all multiples of 5. From each of them, we can get 1 zero; They * * * have six numbers and can get six zeros.

Just six zeros? Will there be more?

Whether it can be more or not depends on the number of prime factors 5. 25 is the square of 5, including two prime factors 5, and 1 5. From 1 multiplied by 30, although only 6 of the 30 factors are multiples of 5, it contains 7 prime factors 5. So the product has seven zeros at the end.

Anyone who multiplies by 30 will do it, no matter how big it is.

For example, multiply this time, from 1 to 100:

1×2×3×4×…×99× 100。 How many zeros are there at the end of the product now?

The answer is 24.