# units digit of the product (3^75)(2^113)

• January 8th 2011, 01:28 PM
bigwave
units digit of the product (3^75)(2^113)
Q1. What is the units digit of the product $(3^{75})(2^{113})$

was not sure of what "units digit" meant but this comes out to be

$
6.31 × 10^{69}
$

otherwise how is this found?

Q2. The product of three different positive intergers is 2010. What is the maximum possible sum of the intergers?
• January 8th 2011, 01:30 PM
dwsmith
Solve mod 10

$3^{75}*2^{113}\equiv x \ \mbox{(mod 10)}$

$3^4\equiv 1 \ \mbox{(mod 10)}$

$3^{72+3}=(3^4)^{18}*3^3\equiv 1^{18}*3^3\equiv 7 \ \mbox{(mod 10)}$

Get the idea?
• January 8th 2011, 02:23 PM
Also sprach Zarathustra
hint for q2:

2010=2*3*5*67
• January 8th 2011, 02:41 PM
Also sprach Zarathustra
for q1:

better and easy to compute 6^75(mod10)

and the we left with: 2^38(mod10)=4^6*2^2=2^14=8^2=4(mod10)