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?
+33614 "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.
