Question

What is the greatest positive power of 5 that divides 1357! exactly with reminders?

Answer 1

Hey.

Here is your answer.


We know that any factorial (!) can be broken into product of consecutive prime numbers with distinct real positive powers.

Like N ! = (2^p)×(3^q)×(5^r)......(X^z)

where 2,3,5....X are consecutive prime factors
and p,q,r...z are the respective powers.


Here we have to find the greatest positive power of 5 i.e., r (here).

so, r can be written as

[N/5] + [N/5^2] + [N/5^3]+............

where [. ] indicates Greatest Integer Function .

Here we have N = 1357


so, we got

r =

[1357/5] + [1357/5^2] + [1357/5^3] + [1357/5^4] + [1357/5^5] ..........


so, r = 271 + 54 + 10 + 2 + 0 + 0 +......

as G.I.F for values less than 1 comes to be 0. do all further values will come to be 0.


so, r = 337.



Hence, 337 is the required greatest power of 5 which will divide 1357 ! exactly.


Thanks

Answer 2

Answer:

hence, we conclude that the greatest power of 5 is 4 which divides 1357 and leaves aremainder 107.

Step-by-step explanation:

so as per the given question, we have to find a power that is positive, for 5 such that it divides 1357 and leaves a remainder, and hence we have to calculate the remainder

first of all we will see different powers of 5 ,

5²=25

5³= 5 x 5 x 5 = 125

$5^4=5*5*5*5=625$

we repeatedly multiply the powers for getting successive powers of 5.

every power of 5 divides the given number 1357 but the greatest one to divide is $5^4$= 625

now we will divide 1357 by 625

$\frac{1357}{625}$

when we divide it

we will get quotient as as 2 as 625x2=1250

and on subtracting it from 1357 we get 107 as remainder.

hence, we conclude that the greatest power of 5 is 4 which divides 1357 and leaves a remainder 107.

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