Bunuel wrote:
Mary chose an even 4-digit number n. She wrote down all the divisors of n in increasing order from left to right: 1, 2, ..., n/2, n. At some moment Mary wrote 323 as a divisor of n. What is the smallest possible value of the next divisor of n written to the right of 323?
(A) 324
(B) 330
(C) 340
(D) 361
(E) 646
Project PS Butler
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Subscribe via RSS - RSS Are You Up For the Challenge: 700 Level QuestionsCall the 4 digit number, N.
We know 323*some factor K1 = N
We want the minimum number X above 323 that multiplied by its factor K2 also equals N.
So N=323K1 = X*K2
We want to minimize X and maximize K2.
Let K2 then equal K1 minus an increment L. We want to minimize L.
So 323K1 = X(K1-L), so
X= 323K1/(K1-L). 323 = 17*19 so
17*19K1/(K1-L) = X
If we want to assume L=1 for the sake of minimization, K1 can't be divisible by (K1-1) unless K1=2, which it can't be because K1 has to be at least 4 in order to create a 4 digit number.
So that leaves 17 or 19 to be divisible by K1-1. So
K1= 18 or 20. Trying both
19*17*18/17 = 19*18 = 342
19*17*20/19 = 17*20 = 340
So the minimum X, the next factor higher than 323 is 340
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