There are 180 positive integers less than or equal to 297 that are relatively prime to 297. To calculate this, the problem breaks down the numbers into those that are multiples of 11, 3, and 33 (the prime factors of 297) and uses the inclusion-exclusion principle to account for overlapping multiples. It finds there are 27 multiples of 11, 99 multiples of 3, and 9 multiples of 33, and subtracting the overlapping ones gives the total of 180 numbers relatively prime to 297.
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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/