QuestionWhat is the minimum number of powers of 2 prizes that can sum to $\$ 818$ ?

Studdy Solution

STEP 1

Assumptions1. The prizes are powers of. . The total prize amount is $818.
3. The goal is to distribute the total prize amount using the minimum number of prizes.

STEP 2

First, we need to find the largest power of2 that is less than or equal to $818. This will be the value of the largest prize.To do this, we can use the formula for the nth power of2, which is$ 2^n $, and find the highest value of n such that$ 2^n \leq818$.

STEP 3

By trial and error, we find that $2^{9} =512$ and $2^{10} =102$ . Therefore, the largest power of2 less than or equal to818 is $2^{9}$ .

STEP 4

Now, subtract the largest power of2 from the total prize amount to find the remaining amount to be distributed.
$Remaining\, amount = Total\, prize\, amount - Largest\, power\, of\,2$

STEP 5

Plug in the values for the total prize amount and the largest power of2 to calculate the remaining amount.
$Remaining\, amount = \$818 - \$512$

STEP 6

Calculate the remaining amount.
$Remaining\, amount = \$818 - \$512 = \$306$

STEP 7

Repeat steps3 to6 until the remaining amount is0. Each time, find the largest power of2 that is less than or equal to the remaining amount, subtract it from the remaining amount, and count the number of prizes.
For the remaining amount of $306, the largest power of2 is$ 2^{} =256$.

STEP 8

Calculate the new remaining amount.
$Remaining\, amount = \$306 - \$256 = \$50$

STEP 9

For the remaining amount of $50, the largest power of2 is$ 2^{5} =32$.

STEP 10

Calculate the new remaining amount.
$Remaining\, amount = \$50 - \$32 = \$18$

STEP 11

For the remaining amount of $18, the largest power of is$ ^{4} =16$.

STEP 12

Calculate the new remaining amount.
$Remaining\, amount = \$18 - \$16 = \$2$

STEP 13

For the remaining amount of $2, the largest power of2 is$ 2^{} =2$.

STEP 14

Calculate the new remaining amount.
$Remaining\, amount = \$2 - \$2 = \$0$

STEP 15

Count the number of prizes, which is the number of times we subtracted a power of2 from the total prize amount.
The minimum number of prizes that could be distributed from $818 is5.

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