2018 AMC 10 B
Complete problem set with solutions and individual problem pages
Mary chose an even digit number . She wrote down all the divisors of in increasing order from left to right: , ,, , . At some moment Mary wrote as a divisor of . What is the smallest possible value of the next divisor written to the right of ? (2018 AMC 10B Problem, Question#21)
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Prime factorizing gives you . The desired answer needs to be a multiple of or , because if it is not a multiple of or , the LCM, or the least possible value for , will not be more than digits. Looking at the answer choices , is the smallest number divisible by or . Checking, we can see that would be .
Let the next largest divisor be . Suppose gcd . Then,as , therefore, . However, because , .
Therefore, gcd . Note that . Therefore, the smallest the gcd can be is and our answer is .
