Number Theory

Now, let's note something really important. 

To determine the units digit of a number, we must focus on the unit digits of the two numbers we are multiplying. 

Any other term actually doesn't matter. 

 

Thus, we have

\(3*7=21\)

 

So the units digit is 1. 

 

Also, we could multiply these two numbers together, since both are small. 

\(13*287=3731\)

 

The units digit is 1. 

 

So our final answer is 1. 

 

Thanks! :)

 

Find the units digit of $13*287$.   

 

13⁰ = 1            13⁴ = 28561            13⁸ = 615730721               13¹² = 23298085122481   

13¹ = 13          13⁵ = 371293          13⁹ = 10604499373           13¹³ = 302875106592253   

13² = 169        13⁶ = 4826809        13¹⁰ = 137858491849       13¹⁴ = 3937376385699289   

13³ = 2197      13⁷ = 62748517      13¹¹ = 1792160394037      13¹⁵ = my calculator went into scientific notation   

 

The above tabulation shows that the units digit repeats every four exponents.   

 

Divide the exponent by 4, disregard the whole number part of the   

quotient, and consider only the remainder part of the quotient.   

Raise 13 to that remainder and see what the units digit is.   

 

Pick a few examples to demonstrate process.   

 

13⁶      6 / 4 = 1 R 2      13² = 169      therefore the units digit of 13⁶ is 9.   

13⁷      7 / 4 = 1 R 3      13³ = 2197    therefore the units digit of 13⁷ is 7.   

13¹²    12 / 4 = 3 R 0     13⁰ = 1         therefore the units digit of 13¹² is 1.   

 

13²⁸⁷   287 / 4 = 71 R 3    13³ = 2197    therefore the units digit of 13²⁸⁷ is 7.   

_.

The answer is \(13*287=3731\)

So \(1\) is the answer.