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1. What is the units digit of the integer value of \(2007^{2008} + 2008^{2007}?\) ?

2. A restaurant mixes 2 gallons of milk containing 1% fat and 3 gallons of milk containing 2% fat. What is the percent of fat in the mixture? Express your answer to the nearest tenth of a percent.

Nov 28, 2020

#1
+114592
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1.  Look at the eding digits  for powers of 7

7^1  =  7

7^2 =   9

7^3  =  3

7^4  =  1

And this pattern repeats  for every  4 powers of 7

So   2008 mod 4  =  0

So....the units digit for  2007 ^2008   =  1

Likewise.....look at the ending digits for  powers  of 8

8^1  =  8

8^2 =  4

8^3  =  2

8^4  =  6

This also repeats for  every  4 powers of 8

So....2007  mod 4  = 3

The units digit for 2008^2007  =  2

So.....the  ending units digit  for 2007^2008 + 2007^2008 =  1 + 2   =  3

Nov 28, 2020
#2
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Nov 28, 2020
edited by Guest  Nov 28, 2020
#3
+114592
+1

2. A restaurant mixes 2 gallons of milk containing 1% fat and 3 gallons of milk containing 2% fat. What is the percent of fat in the mixture? Express your answer to the nearest tenth of a percent.

1%  of 2  =  .02  gallons of fat

2% of 3 =    ,06 gallons of fat

% fat in the mixture  =    [ .02 + .06 \  /  [ 2 + 3 ] =      .08 / 5  = .016  =  1.6%

Nov 28, 2020