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Problem 29


Problem 29


Distinct powers

Consider all integer combinations of ab for 2 ≤ a ≤ 5 and 2 ≤ b ≤ 5:

22=4, 23=8, 24=16, 25=32
32=9, 33=27, 34=81, 35=243
42=16, 43=64, 44=256, 45=1024
52=25, 53=125, 54=625, 55=3125

If they are then placed in numerical order, with any repeats removed, we get the following sequence of 15 distinct terms:

4, 8, 9, 16, 25, 27, 32, 64, 81, 125, 243, 256, 625, 1024, 3125

How many distinct terms are in the sequence generated by ab for 2 ≤ a ≤ 100 and 2 ≤ b ≤ 100?


不同的幂

考虑所有满足2 ≤ a ≤ 5和2 ≤ b ≤ 5的整数组合生成的幂ab

22=4, 23=8, 24=16, 25=32
32=9, 33=27, 34=81, 35=243
42=16, 43=64, 44=256, 45=1024
52=25, 53=125, 54=625, 55=3125

如果把这些幂按照大小排列并去重,我们得到以下由15个不同的项组成的序列:

4, 8, 9, 16, 25, 27, 32, 64, 81, 125, 243, 256, 625, 1024, 3125

在所有满足2 ≤ a ≤ 100和2 ≤ b ≤ 100的整数组合生成的幂ab排列并去重所得到的序列中,有多少个不同的项?