Call an integer $n$ oddly powerful if there exist positive integers $a$ and $b$, where $b>1$, $b$ is odd, and $a^b = n$. How many oddly p

Question

Call an integer $n$ oddly powerful if there exist positive integers $a$ and $b$, where $b>1$, $b$ is odd, and $a^b = n$. How many oddly powerful integers are less than $2010$?

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Ella 3 weeks 2021-09-07T23:19:30+00:00 1 Answer 0

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    2021-09-07T23:20:57+00:00

    Answer:

    There are 16 oddly powerful integers less than 2010

    Step-by-step explanation:

    ∵ b is an odd integer

    ∵ b > 1

    ∴ The first value of b is 3

    ∵ a is an integer

    – We can use a = 1, 2, 3, ……….

    a^{b}=n

    ∵ n < 2010

    – Let a = 1, 2, …………… 12 because 12³ is greatest integer  < 2010

    ∵ 1³ = 1, 2³ = 8, 3³ = 27, 4³ = 64, 5³ = 125, 6³ = 216, 7³ = 343,

       8³ = 512, 9³ = 729, 10³ = 1000, 11³ = 1331, 12³ = 1728

    ∴ There are 12 oddly powerful integers with b = 3

    Now the second value of b is 5

    1^{5}=1 but we took 1 before so we will start with 2

    2^{5}=32, 3^{5}=243, 4^{5}=1024

    4^{5} is the greatest integer < 2010

    ∴ There are 3 oddly powerful integers with b = 5

    Now the third value of b is 7

    2^{7}=128

    2^{7} is the greatest integer < 2010

    ∴ There is 1 oddly powerful integers with b = 7

    Now the fourth value of b is 9

    2^{9}=512

    2^{9} is the greatest integer < 2010

    – But we used 512 before

    ∴ There is no oddly powerful integers with b = 9

    – 9 is the greatest value of b which makes a^{b}<2010

    ∵ 12 + 3 + 1 = 16

    There are 16 oddly powerful integers less than 2010

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