At the outset, I wish to acknowledge the superb analytical prowess of Ady TZIDON whose impeccable reasoning deducing the process as well as solution to the first part of the problem,in my opinion, allowed other members of the site to solve the second part. Credits are also in order for HarryHotspur and Leming for solving the second part.

SOLUTION TO PART A:

The required smallest number is 70,007,000,000.

Explanation:

It can be established by trial and error that the smallest number constituted entirely by ones which is divisible by 137 is 10,001 so that the smallest number constituted entirely by sevens which is divisible by 137*7=959 is 70,007. Now, any number having precisely six zeroes as the last six digits is always divisible by 64. Since, 64 is relatively prime to 959, it follows that the smallest number constituted entirely by sevens and zeroes which is divisible by 959*64 = 61376 is 70,007,000,000.

SOLUTION TO PART B:

The required smallest number is 7,777,777,000,000.

Explanation:

It can be established by trial and error that the smallest number constituted entirely by ones which is divisible by 239 is 1,111,111 so that the smallest number constituted entirely by sevens which is divisible by 239*7=1673 is 7,777,777. Now, any number having precisely six zeroes as the last six digits is always divisible by 64. Since, 64 is relatively prime to 1673, it follows that the smallest number constituted entirely by sevens and zeroes which is divisible by 1673*64 = 107072 is 7,777,777,000,000.

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