AMC 8 Daily Practice Round 6

Complete problem set with solutions and individual problem pages

Problem 30 Easy

The number 2^{48}-1 is divisible by two integers between 60 and 70. What are these integers?

  • A.

    63,65

  • B.

    63,64

  • C.

    64,65

  • D.

    65,66

  • E.

    67,68

Answer:A

2^{48}-1 = (2^{24}-1)(2^{24}+1) \\

= (2^{12}-1)(2^{12}+1)(2^{24}+1) \\

= (2^6-1)(2^6+1)(2^{12}+1)(2^{24}+1) \\

= 63 \times 65 \times (2^{12}+1)(2^{24}+1)

The factors between 60-70 are 63 and 65.

Final results: \boxed{63,65}

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