1)Find the largest integer n for which 12^n evenly divides 20! .

2)Let A be the product of the divisors of 300. What is the sum of the distinct prime divisors of A?

3)What is the sum of all positive integers n that satisfy

lcm[n,100]=gcd(n,100)+450

DragonXlayer Sep 23, 2020

#1**0 **

4)In the figure below, the largest circle has a radius of six meters. Five congruent smaller circles are placed as shown and are lined up in east-to-west and north-to-south orientations. What is the radius in meters of one of the five smaller circles?

IMAGE:

https://latex.artofproblemsolving.com/e/d/2/ed239c0851978d4e1f82ba24458bd2de9836d085.png

DragonXlayer Sep 24, 2020

#2**+1 **

1 - 20! mod 12^8 = 0

2 - product(1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 25, 30, 50, 60, 75, 100, 150, 300)=19683000000000000000000

19683000000000000000000 = 2^18 * 3^9 * 5^18

Sum = 2 + 3 + 5 = 10

3 - I could find only one value for n:

n = 250

Guest Sep 24, 2020

#3**0 **

thanks, for number one if n is 0 then 12^0=1 and 1 does divde into 20! but is that realy the largest? (just asking)

DragonXlayer
Sep 24, 2020