An arithmetic sequence with first term 1 has a common difference of 6. A second sequence begins with 4 and has a common difference of 7. In the range of 1 to 100, what is the largest number common to both sequences?

Guest Aug 9, 2018

#1**+1 **

"*An arithmetic sequence with first term 1 has a common difference of 6. A second sequence begins with 4 and has a common difference of 7. In the range of 1 to 100, what is the largest number common to both sequences?*"

We must have:

1 + 6n = 4 + 7m where m and n are integers

This can be rearranged as:

6n - 7m = 3 or

2n - 7m/3 = 1

m must be a multiple of 3, and, because 2n is always even, 7m/3 must be odd to give a difference of 1. This means m must be an odd multiple of 3.

m n 2n-7m/3 4+7m 1+6n

3 4 1 25 25

9 11 1 67 67

15 18 1 109 109

So 67 is the largest number in the range 1 to 100 that is common to both sequences.

Alan Aug 9, 2018