Problem of the Week 28

Find a, b, c, d, e, and f given the following conditions:

a (b + c + d + e + f) = 184
b (a + c + d + e + f) = 225
c (a + b + d + e + f) = 301
d (a + b + c + e + f) = 369
e (a + b + c + d + f) = 400
f (a + b + c + d + e) = 525

 

Alternate Problem 28

A Fibonacci sequence is formed by starting with any two integers, then adding them to get the next term.  Any term thereafter is obtained by adding the two preceding terms.  For example, starting with 2 and 5, the following sequence is formed:

2, 5, 7, 12, 19, 31, 50, 81, 131, 212, 343, 555, 898, …

If any ten consecutive terms of a Fibonacci sequence are added, the sum is a constant multiple of one of the ten terms.  Find the constant and the term, relative to the other terms.

 

Extension Problem 28

multiply perfect number is one for which the sum of all  the factors including one and the number itself is a multiple of the given number.  For example, 30,240 is a multiply perfect number.  What multiple of 30,240 gives the sum of all its factors?