Question

The least value of positive integer n , for which 1000!/13^n is not an integer .

Answer 1

@ganeshie8

Answer 2

find the exponent of 13 in prime factorization of 1000!

Answer 3

How ?

Answer 4

well, how many times do you hit a multiple of 13 when going from 1 to 1000?

Answer 5

Because a factorial is just a product of all consecutive numbers.

Answer 6

Also consider squares of \(13\) as well

Answer 7

Consider the following:
\[\text{ does } 13|1000! \\ \text{ does } 13^2|1000! \\ \text{ does } 13^3=13 \cdot 13^2 |1000! \\ \text{ does } 13^4=13 \cdot 13^2 \cdot 13 |1000! \\ \text{ does } 13^5=13 \cdot 13^2 \cdot 13^2 |1000! \\ \text{ does } 13^6=13 \cdot 13^2 \cdot 13^3 |1000! \\ \text{ blah blah .... }\]
like think about the first one we know 1*2*3*4*5*6*7*8*9*10*11*12*13*....*1000 is 1000!
and there is a 13
so 13|1000!

think about the second one 13^2=169 and 169 is less than 1000 so 169 is in 1000!

think about the third one 13 and 169 is in 1000!

think about the fourth one 13 and 169 and 13 again is 1000!
that other 13 is in 1000! because 13*2=26

think about the others...

Answer 8

umm..what do you mean by 13|1000!

Answer 9

a|b means a divides b
ak=b for some integer k
a is a factor of b

Answer 10

for example :

2 | 6

5 | 25

3 | 6

Answer 11

right
2(3)=6 so 2|6

but
there is no integer such that 2n=11 so saying 2|11 is not true
\[2 \cancel{ | }11 \]

Answer 12

okay...thinking over what you said..

Answer 13

yes..understood

Answer 14

are you able to figure out the value of \(n\) ? :)