The number of terms common to both the arithmetic progressions 2,5,8,11,...., 179 and 3,5,7,9,....., 101 is

The first common term in both the arithmetic progressions is 5. The common difference will be the LCM of the common difference of the two AP. Thus, the common difference = LCM(2,3) = 6

Also, the last common term should be less than or equal to 101.

=> $$a=5$$, $$d=6$$, and $$a_n\le101$$

=> $$a_n\le101$$

=> $$a+(n-1)d\le101$$

=> $$5+(n-1)6\le101$$

=> $$(n-1)6\le96$$

=> $$(n-1)\le16$$

=> $$n\le17$$

Thus, the number of common terms is 17.