Question 60968: find the units digit of 3^13*7^14 without actual multiplication
Answer by joyofmath(189)   (Show Source):
You can put this solution on YOUR website! We're looking at 
Well,  so  ends in 3.
 ends in 3*3 or 9.
 ends in 9*3 =27 so it ends in 7.
 ends in 7*3 = 21 so it ends in 1.
 ends in 1*3 or 3. So it ends in the same as .
 ends in 3*3 ends the same as .
 ends in 9*3 ends the same as .
So, the pattern repeats every 4 powers. So,  ands with the same digit as  and which is 3.
 ends in 7.
 ends in the same digit as 7*7, or 9.
 ends in the same digit as 9*7, or 3.
 ends in the same digit as 3*7, or 1.
 ends in the same digit as 1*7, which ends the same way as .
So, this pattern also repeats every 4 powers.
So,  ends the same way as  and  and which is with a 9.
So, ends with the same digit as which ends in 3 multiplied by the last digit of which is 9.
So, the big product ends in  so 7 is the last digit of the big product.
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