2016 AMC 12B Problems/Problem 24
There are exactly ordered quadruplets such that and . What is the smallest possible value for ?
Let , etc., so that . Then for each prime power in the prime factorization of , at least one of the prime factorizations of has , at least one has , and all must have with .
Let be the number of ordered quadruplets of integers such that for all , the largest is , and the smallest is . Then for the prime factorization we must have So let's take a look at the function by counting the quadruplets we just mentioned.
There are quadruplets which consist only of and . Then there are quadruplets which include three different values, and with four. Thus and the first few values from onwards are Straight away we notice that , so the prime factorization of can use the exponents . To make it as small as possible, assign the larger exponents to smaller primes. The result is , so which is answer .
Also, to get the above formula of , we can also use the complementary counting by doing , while the first term is for the four integers to independently have choices each, with the second term indicating to subtract all the possibilities for the four integers to have values between and , and similarly the third term indicating to subtract all the possibilities for the four integers to have values between and , in the end the fourth term meaning the make up for the values between and .
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