If the total number of
-digit positive integers that can be formed using the digits
to
inclusive and without repetition. such that
always exist collectively in the same order in the positive integer is
.
We coin the term: Cute divisors . A cute divisor of a number is a positive integer other than and which divide the number completely and leaves no remainder.
Find the sum of all cute divisors of .
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Since 1 2 3 4 must exist contiguously, we can consider them as a single digit A .
Since the original number is 8 digit, the number now reduces to 5 digit numbers ( A 4 A 3 A 2 A 1 A 0 ) that can be formed using digits { 0 , A , 5 , 6 , 7 , 8 , 9 } .
Total ways to place these 7 digits in a 5 digit number, N ( T ) = 7 × 6 × 5 × 4 × 3 = 2 5 2 0
Total 5 digit numbers that had A 4 as 0 , N ( Z ) = 1 × 6 × 5 × 4 × 3 = 3 6 0
Total 5 digit numbers that did not have A , N ( A ) = 6 × 5 × 4 × 3 × 2 = 7 2 0
Total 5 digit numbers that did not have A and had A 4 as 0 , N ( A ∩ Z ) = 1 × 5 × 4 × 3 × 2 = 1 2 0
∴ Total valid 5 digit numbers that have A , P
= N ( T ) − N ( Z ) − N ( A ) + N ( A ∩ Z )
= 2 5 2 0 − 3 6 0 − 7 2 0 + 1 2 0
= 1 5 6 0
Prime factorization of P = 2 3 × 3 × 5 × 1 3
Sum of divisors of P = 2 − 1 2 4 − 1 × ( 3 + 1 ) × ( 5 + 1 ) × ( 1 3 + 1 ) = 5 0 4 0
Sum of cute divisors of P = 5 0 4 0 − 1 − 1 5 6 0 = 3 4 7 9