[求助]请教NN两个DS难题

chengcheng7

请教NN两个DS的解题思路

1.      If M, N, P are three different prime integers and X, Y, Z are three positive integers, is (M)^x (N)^y (P)^z > 100?

(1)   x + y + z = 5

(2)   M × N × P = 30

答案是A

2.      If K is the product of the first 30 positive integers and d is a positive integer, what is the value of d?

 

(1)   (10)^d is a factor of K

(2)    d > 6

答案是C

johnsonwxjs

1.      If M, N, P are three different prime integers and X, Y, Z are three positive integers, is (M)^x (N)^y (P)^z > 100?

From 1) ==> X, Y, Z are 1, 2, 2  =>Smallest (M)^x (N)^y (P)^z  = (2)^x (3)^y (5)^z =  (2)^2 (3)^{^} 2 (5)^1 =180 >100;

                                     or 1, 1, 3  ==>Smallest (M)^x (N)^y (P)^z  = (2)^x (3)^y (5)^z =  (2)^3 (3)^{^} 1 (5)^1 =120 >100;

From 2) ==> M, N, P are 2, 3, 5

                      If X=Y=Z=1 => (M)^x (N)^y (P)^z  = (2)^x (3)^y (5)^z =  (2)^1 (3)^{^} 1 (5)^1 =30 <100

                       If X=Y=Z=10 => (M)^x (N)^y (P)^z  = (2)^x (3)^y (5)^z >100

   

2.      If K is the product of the first 30 positive integers and d is a positive integer, what is the value of d?

 

(1)   (10)^d is a factor of K

(2)    d > 6

Since  K= 30! , from (1) we know that d could equal to 1, 2, 3, or more. because K's factor includes at least 10, 20, 30.

(2) alone just not eough;

For (1)+(2) ==>  30! = X1*(5x10x15x20x25x30) =X2* (10^7)  

              note: we can borrow enough 2 from X1 to alter 5 to 10; 25 = 5x5

From (1)+ (2) ==>The smallest d is 7.
 ==> d=7 answer is C

智者多能

 

Great, thank you very much. Only one thing,

QUOTE:

For (1)+(2) ==>  30! = X1*(5x10x15x20x25x30) =X2* (10^7)  

              note: we can borrow enough 2 from X1 to alter 5 to 10; 25 = 5x5

From (1)+ (2) ==>The smallest d is 7.
 ==> d=7 answer is C,

              note: we can borrow enough 2 from X1 to alter 5 to 10; 25 = 5x5

From (1)+ (2) ==>The smallest d is 7.
 ==> d=7 answer is C,

K=30!, from (1)==>the range of possible d is from 1~7, then (1)+(2) suggests the answer should be the biggest d, 7.

If I have any misunderstanding, please correct me. Thanks.

chengcheng7

感谢!!!

果然是数学NN!!!