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

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, ……….

∵ 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

, ,

is the greatest integer < 2010

∴ There are 3 oddly powerful integers with b = 5

Now the third value of b is 7

is the greatest integer < 2010

∴ There is 1 oddly powerful integers with b = 7

Now the fourth value of b is 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

∵ 12 + 3 + 1 = 16

There are 16 oddly powerful integers less than 2010