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def seven_zero(n): Seven is considered a lucky number in Western cultures, whereas zero is what nobody wants to be. Let us briefly bring these two opposites together by looking at positive integers that consist of some solid sequence of sevens, followed by some (possibly empty) solid sequence of zeros. Examples of integers of this form are 7, 77777, 7700000, 77777700, and 70000000000000. A surprising theorem proves that for any positive integer n, there exist infinitely many integers of such seven- zero form that are divisible by n. This function should return the smallest such seven-zero integer. Even though discrete math and number theory help, this exercise is not about coming up with a clever symbolic formula and the proof of its correctness. This is rather about iterating through the numbers of this constrained form of sevens and zeros efficiently and correctly in strictly ascending order, so that the function can mechanistically find the smallest working number of this form. This logic might be best written as a generator to yield such numbers. The body of this generator consists of two nested loops. The outer loop iterates through the number of digits d in the current number. For each d, the inner loop iterates through all possible k from one to d to create a number that begins with a block of k sevens, followed by a block of d-k zeros. Most of its work done inside that helper generator, the seven_zero function itself will be short and sweet. Expected result 17 רר7ררררד77ר77777

def seven_zero(n): Seven is considered a lucky number in Western cultures, whereas zero is what nobody wants to be. Let us briefly bring these two opposites together by looking at positive integers that consist of some solid sequence of sevens, followed by some (possibly empty) solid sequence of zeros. Examples of integers of this form are 7, 77777, 7700000, 77777700, and 70000000000000. A surprising theorem proves that for any positive integer n, there exist infinitely many integers of such seven- zero form that are divisible by n. This function should return the smallest such seven-zero integer. Even though discrete math and number theory help, this exercise is not about coming up with a clever symbolic formula and the proof of its correctness. This is rather about iterating through the numbers of this constrained form of sevens and zeros efficiently and correctly in strictly ascending order, so that the function can mechanistically find the smallest working number of this form. This logic might be best written as a generator to yield such numbers. The body of this generator consists of two nested loops. The outer loop iterates through the number of digits d in the current number. For each d, the inner loop iterates through all possible k from one to d to create a number that begins with a block of k sevens, followed by a block of d-k zeros. Most of its work done inside that helper generator, the seven_zero function itself will be short and sweet. Expected result 17 רר7ררררד77ר77777

C++ for Engineers and Scientists
C++ for Engineers and Scientists
4th Edition
ISBN:9781133187844
Author:Bronson, Gary J.
Publisher:Bronson, Gary J.
Chapter4: Selection Structures
Section: Chapter Questions
Problem 14PP
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Sevens rule, zeros drool def seven_zero(n): Seven is considered a lucky number in Western cultures, whereas zero is what nobody wants to be. Let us briefly bring these two opposites together by looking at positive integers that consist of some solid sequence of sevens, followed by some (possibly empty) solid sequence of zeros. Examples integers of this form are 7, 77777, 7700000, 77777700, and 70000000000000. A surprising theorem proves that for any positive integer n, there exist infinitely many integers of such seven- zero form that are divisible by n. This function should return the smallest such seven-zero integer. Even though discrete math and number theory help, this exercise is not about coming up with a clever symbolic formula and the proof of its correctness. This is rather about iterating through the numbers of this constrained form of sevens and zeros efficiently and correctly in strictly ascending order, so that the function can mechanistically find the smallest working number of this form. This logic might be best written as a generator to yield such numbers. The body of this generator consists of two nested loops. The outer loop iterates through the number of digits d in the current number. For each d, the inner loop iterates through all possible k from one to d to create a number that begins with a block of k sevens, followed by a block of d-k zeros. Most of its work done inside that helper generator, the seven_zero function itself will be short and sweet. Expected result n 17 ר77ך77ררררררךררר 42 0ררר 103 דדרררר777 ר777777ררךררו 77700 77700 2**50 700000000000000000000000000000000000000000000000000 12345 (a behemoth that consists of 822 sevens, followed by a single zero) This problem is adapted from the excellent MIT Open Courseware online textbook "Mathematics for Computer Science" (PDF link to the 2018 version for anybody interested) that, like so many other non-constructive combinatorial proofs, uses the pigeonhole principle to prove that some solution must exist for any integer n, but provides no clue about where to actually find that solution. Also as proven in that same textbook, whenever n is not divisible by either 2 or 5, the smallest such number will always consist of some solid sequence of sevens with no zero digits after them. This can speed up the search by an order of magnitude for such friendly values of n.
Transcribed Image Text:Sevens rule, zeros drool def seven_zero(n): Seven is considered a lucky number in Western cultures, whereas zero is what nobody wants to be. Let us briefly bring these two opposites together by looking at positive integers that consist of some solid sequence of sevens, followed by some (possibly empty) solid sequence of zeros. Examples integers of this form are 7, 77777, 7700000, 77777700, and 70000000000000. A surprising theorem proves that for any positive integer n, there exist infinitely many integers of such seven- zero form that are divisible by n. This function should return the smallest such seven-zero integer. Even though discrete math and number theory help, this exercise is not about coming up with a clever symbolic formula and the proof of its correctness. This is rather about iterating through the numbers of this constrained form of sevens and zeros efficiently and correctly in strictly ascending order, so that the function can mechanistically find the smallest working number of this form. This logic might be best written as a generator to yield such numbers. The body of this generator consists of two nested loops. The outer loop iterates through the number of digits d in the current number. For each d, the inner loop iterates through all possible k from one to d to create a number that begins with a block of k sevens, followed by a block of d-k zeros. Most of its work done inside that helper generator, the seven_zero function itself will be short and sweet. Expected result n 17 ר77ך77ררררררךררר 42 0ררר 103 דדרררר777 ר777777ררךררו 77700 77700 2**50 700000000000000000000000000000000000000000000000000 12345 (a behemoth that consists of 822 sevens, followed by a single zero) This problem is adapted from the excellent MIT Open Courseware online textbook "Mathematics for Computer Science" (PDF link to the 2018 version for anybody interested) that, like so many other non-constructive combinatorial proofs, uses the pigeonhole principle to prove that some solution must exist for any integer n, but provides no clue about where to actually find that solution. Also as proven in that same textbook, whenever n is not divisible by either 2 or 5, the smallest such number will always consist of some solid sequence of sevens with no zero digits after them. This can speed up the search by an order of magnitude for such friendly values of n.
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