AMC 10 Daily Practice Round 2

Complete problem set with solutions and individual problem pages

Problem 11 Medium

Consider positive integers a,b,c,d such that ab=cd, which of the following is a possible value of a+b+c+d?

  • A.

    97

  • B.

    101

  • C.

    301

  • D.

    401

  • E.

    None of the above

Answer:C

Since 97, 101, and 401 are both prime and 301=7\times 43.

Let a=mn, b=pq, c=mp, d=nq, m, n, p, q\in {{\mathbb{N}}^{*}},

Then a+b+c+d=mn+pq+mp+nq=(m+q)(n+p)，

So we are good as long as a+b+c+d is not prime.

For example, 301=7\times 43=(1+6)\cdot (1+42).

This is true when a=1, b=252, c=42, d=6.

Related Problems
Problem 9 AMC 10 Daily Practice Round 2
Problem 10 AMC 10 Daily Practice Round 2
Problem 12 AMC 10 Daily Practice Round 2
Problem 13 AMC 10 Daily Practice Round 2
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