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# Number Theory

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Find the units digit of \$13*287\$.

Jul 9, 2024

#1
+1256
+1

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! :)

Jul 9, 2024
#3
+944
-1

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.

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Jul 10, 2024
#4
+174
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