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Solution
How many positive integers less than or equal to are relatively prime to ?297 297
Since , a multiple of neither nor is relatively prime to . Let denote the positive integers less than or equal to
that are multiple of a positive integer , then
Moreover, since the number of the common multiples of and is the same as that of the multiple of which is the least common multiple of
and , .
Thus, the number of positive integers that are multiple of either or is
Therefore, the number to be calculated is .
297 = 11 × 33
11 3 297 n(p) 297
p
n(11) = 297/11 = 27,n(3) = 297/3 = 99.
11 3 33
11 3 n(33) = 297/33 = 9
11 3
n(11) + n(3) − n(33) = 27 + 99 − 9 = 117.
297 − 117 = 180
Many relatively prime numbers
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6/21/2013https://brilliant.org/assessment/kt/solvable_component/gcd-lcm/1960807/