to solve the problem, we need to find the number of positive integers (n) such that:
- (n) is a multiple of 7, and
- (n) has exactly 7 positive divrs.
key observations:
the number of positive divrs of a number (n) (denoted (\tau(n))) is determined by its prime factorization. for (n = p_1^{k_1}p_2^{k_2}\dots p_m^{k_m}), (\tau(n) = (k_1 1)(k_2 1)\dots(k_m 1)).
since (\tau(n) = 7) (a prime number), the only way this product equals 7 is if there is exactly one prime factor with exponent (6) (because (6 1=7)). thus, (n = p^6) for some prime (p).
condition for being a multiple of 7:
(n = p^6) must be a multiple of 7. since 7 is prime, (p) must be 7.
thus, the only such (n) is (7^6 = 117649).
answer: (\boxed{1})
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