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2013 AMC 8

Problem 15

Problem 15 · 2013 AMC 8 Medium
Algebra & Patterns powers-by-inspection

If 3p + 34 = 90, 2r + 44 = 76, and 53 + 6s = 1421, what is the product of p, r, and s?

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Answer: B — 40.
Show hints
Hint 1 of 2
Each equation has one known power and one unknown power. Move the known number to the right side first, and you're left with "(base)? = (a number)" — just ask which power of that base it is.
Still stuck? Show hint 2 →
Hint 2 of 2
To solve bx = N by hand, rewrite N as a power of b and read off the exponent. Recognizing small powers (32=9, 25=32, 64=1296) is the whole skill here.
Show solution
Approach: isolate each power, then recognize it
  1. Equation 1: 3p = 90 − 34 = 90 − 81 = 9 = 32, so p = 2.
  2. Equation 2: 2r = 76 − 44 = 32 = 25, so r = 5.
  3. Equation 3: 6s = 1421 − 53 = 1421 − 125 = 1296 = 64, so s = 4.
  4. Product = 2 · 5 · 4 = 40.
  5. Tip: when an exponential equation has a clean integer answer, the leftover number is almost always a recognizable power of the base — match by eye instead of guessing.
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