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Need help with an AMC8 mock question

 
 
Song Ashley的相片
Need help with an AMC8 mock question
Song Ashley發表於2017年 10月 22日(Sun) 16:32
 

Let wwxxyy, and zz be whole numbers. If 2^w3^x5^y7^z=5882w⋅3x⋅5y⋅7z=588

then what does 2w+3x+5y+7z2w+3x+5y+7z equal?

(A) 21(B) 25(C) 27(D) 35(E) 56(A) 21(B) 25(C) 27(D) 35(E) 56

Could someone please help explain how to solve this problem? THanks!

 
Song Ashley的相片
Re: Need help with an AMC8 mock question
Song Ashley發表於2017年 10月 22日(Sun) 16:33
 

When I tried to copy and past, it looks normal, but them when I saved it, it is duplicated and formatted weird. 

Institute Areteem的相片
Re: Need help with an AMC8 mock question
Institute Areteem發表於2017年 10月 26日(Thu) 10:19
 

What have you tried so far Ashley?

One good place to start is noticing that the powers on the left are all prime numbers. So, how does $588$ look like when you factor it out using prime numbers?

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Re: Need help with an AMC8 mock question
Liu Anna發表於2017年 10月 27日(Fri) 11:54
 

2, 3, 5, and 7 are prime numbers. The prime factorization of 588 is 2^2 x 3 x 7^2. 3= 3^1, and since there is no five, you write it as 5^0. 2(2) + 3(1) + 5(0) + 7(2) = 4 + 3 + 0 + 14 = 21, so the answer would be (a) 21.