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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?

Aug 9, 2018

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"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.

Aug 9, 2018