Find the number of zeroes

[mathdaemon]

Hello

Find the number of zeroes in the following expression
15*32*25*22*40*75*98*112*125.

I got the ans by prime factorizing all the terms and taking min(number of factors that are 2, number of factors that are 5).
Ans I got is 9.
Please explain the logic to me.

Thank You

 

[stapel]

Find the number of zeroes in the following expression
15*32*25*22*40*75*98*112*125.

I got the ans by prime factorizing all the terms and taking min(number of factors that are 2, number of factors that are 5).
Ans I got is 9.
Please explain the logic to me.

Um... I'm confused. Are you asking us to explain to you your own logic by which you developed your solution (listed above)? :shock:

 

[pka]

Find the number of zeroes in the following expression
15*32*25*22*40*75*98*112*125.
I got the ans by prime factorizing all the terms and taking min(number of factors that are 2, number of factors that are 5).
Ans I got is 9.

I think that you have missquoted this problem.
15*32*25*22*40*75*98*112*125. we can see the factorization here. BUT it is multiplied out here.
You can see that there are actually ten zeros in the answer but there are nine zeros at the end of the number.

Those nine zeros are called trailing zeros, That is a classic question in number theory/counting theory.
We look at the prime factorization $\displaystyle 2^a\cdot 3^b\cdot 5^c\cdot\cdots P^k$. That number has $\displaystyle \min\{a,c\}$ trailing zeros.

Therefore, in your number $\displaystyle \min(14,9)=9$ so there are nine trailing zeros.
If at all possible, working from a factored form it is difficult to see how many total zeros there are.

 

[mathdaemon]

@stapel : Actually, after encountering this question, I googled for the logic but could not find any explanation as to why this works.

@pka : I guess the logic is, minimum number of 2X5 multiplications. Please correct me if I am wrong else THANK YOU.

 

[stapel]

@stapel : Actually, after encountering this question, I googled for the logic but could not find any explanation as to why this works.

Does this mean that you were just applying a formula that they'd given you, and you're asking now for the logic of it? If so, then try the explanation to a related question, being the number of trailing zeroes in the expansion of a factorial.

 

[Deleted member 4993]

@stapel : Actually, after encountering this question, I googled for the logic but could not find any explanation as to why this works.

@pka : I guess the logic is, minimum number of 2X5 multiplications. Please correct me if I am wrong else THANK YOU.

No... it is the maximum number of 2x5 (= 10) multiplications....

 

[ysr1991]

Answer

Hello



Find the number of zeroes in the following expression
15*32*25*22*40*75*98*112*125.

I got the ans by prime factorizing all the terms and taking min(number of factors that are 2, number of factors that are 5).
Ans I got is 9.
Please explain the logic to me.





Thank You

as number of zeros depend on 10. and 10=2x5. so we need to find number of 2's and number of 5's
3x5 x 2x2x2x2x2 x 5x5 x 11x2 x 5x2x2x2 x 5x5x3 x 7x7x2 2x2x2x2x7 x 5x5x5

so number of 5's =9
number of 2's = 14

we can make 9 pairs so number of zeros will be 9

 

[pka]

15*32*25*22*40*75*98*112*125
=(15*32)*(25*22)*(40)*(75*98)*(112*125)
=480 * 550 * 40 * 7350 * 14000
= 264000 * 40 * 102900000
= 264* 100 * 4 * 10 * 1029 * 100000
= 10866240000000
So therefore, there are 9 zeroes

Look at this webpage. The product is 1086624000000000.
I count ten zeros,

 

[mathdaemon]

Trailing 0's.