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Numbers and Their Application to Math and Science

Odd Solutions for HW Numbers Lesson 3

  1. What is the sum of the divisors of 24•(25 - 1) and 26•(27 - 1)? Are these numbers perfect?

    Answer: a) 496, yes    b) 8128, yes
    496 = 16 • 31 = 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248
    8128 = 64 • 127 = 1 + 2 + 4 + 8 + 16 + 32 + 64 + 127 + 254 + 508 + 1016 + 2032 + 4064

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  3. For the number 284, find all the factors; add the factors (except itself); count all the factors; find the prime factorization.

    284 = 22•711, thus 1 + 2 + 4 + 71 + 142 = 220.
    The number of factors is 6. 6 = 3•2 (Again, the pi product of one more than each exponent.)
    284 is called a deficient number. The sum of the divisors is less than itself.
    220 and 284 are called amicable numbers (meaning friendly number) and are the smallest pair. Starting with 12496, you can form a longer chain of sociable numbers.

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  5. 13! (6,227,020,800) OR 14! (8.71782912•1010

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  7. 1, 11, 121, 1331, 14641, 161051, 1771561. Note how the digits refer to Pascal's Triangle!! 161051 is generated by carries out of 1 5 10 10 5 1, since our "digits" would exceed nine.

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  9. 1001 = 7•11•13.

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  11. Write out the prime factorization for both.
    1. Find the GCF(156,182).
      A.156 = 3×22×13; 182 = 2×7×13; so that GCF(156,182)=26.
    2. Find the LCM(496,8128).
      A. The prime factorization of 496 is 24×31, and 8128 is 26×127 so that the LCM(496,8128) = 251,968.

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  13. A. The least common multiple of 6 and 7 is 42, so 6 weeks.
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