Evaluate the function at without using a calculator.
B)-3
C)-1
D)25
E)-2

Question

Evaluate the function  at  without using a calculator.
B)-3 
C)-1 
D)25 
E)-2

Quizplus · Accepted Answer

This question involves performing modular arithmetic. The expression given is in hexadecimal representation, so we need to convert it to decimal by multiplying each digit by its corresponding power of 16 and summing the results. We have:

11eacce6_2010_d5c1_9f49_f77b5edb6fde_TB7896_11 = 
1 × 16^35 + 1 × 16^34 + 14 × 16^33 + 10 × 16^32 + 12 × 16^31 + 
12 × 16^30 + 14 × 16^29 + 6 × 16^28 + 32 × 16^24 + 13 × 16^23 + 
5 × 16^22 + 12 × 16^21 + 1 × 16^20 + 9 × 16^19 + 15 × 16^18 + 
4 × 16^17 + 9 × 16^16 + 247 × 16^15 + 181 × 16^14 + 117 × 16^13 + 
93 × 16^12 + 221 × 16^11 + 111 × 16^10 + 222 × 16^9 + 
246 × 16^8 + 75 × 16^7 + 215 × 16^5 + 185 × 16^4 + 205 × 16^3 + 
214 × 16^2 + 182 × 16^1 + 150

We can simplify this expression using modular arithmetic to avoid large computations. Note that 16 ≡ -2 (mod 18), since 16 - (-2) = 18. Therefore, we can replace 16 with -2 wherever it appears in the expression, and reduce all coefficients modulo 18. We have:

11eacce6_2010_d5c1_9f49_f77b5edb6fde_TB7896_11 ≡ 
1 × (-2)^3 + 1 × (-2)^2 + 14 × (-2) + 10 × 1 + 12 × (-2) + 
12 × 1 + 14 × (-2) + 6 × 1 + 14 × 1 + 13 × (-2) + 
5 × 1 + 12 × (-2) + 1 × 1 + 9 × (-2) + 15 × 1 + 
4 × (-2) + 9 × 1 + 5 × (-2) + 13 × (-2) + 5 × (-2) + 
3 × (-2) + 15 × 1 + 15 × (-2) + 9 × 1 + 13 × 1 + 
13 × (-2) + 6 × 1 + 6 × (-2) + 15 × 1 + 11 × (-2) + 
17 × 1 + 13 × 1 + 16 × (-2) + 8 × (-2) + 14 × (-2) + 
17 × (-2) + 4

Simplifying this expression further, we get:

11eacce6_2010_d5c1_9f49_f77b5edb6fde_TB7896_11 ≡ 
-8 - 2 - 6 + 10 - 4 + 12 - 10 + 6 + 14 - 26 + 5 - 24 - 2 - 18 + 15 - 8 - 9 - 10 - 26 - 10 + 15 - 30 + 9 - 23 - 26 + 6 - 12 + 15 - 22 + 17 + 13 - 16 - 16 - 14 - 17 + 4 (mod 18)

Finally, we can evaluate this expression mod 18 to get:

11eacce6_2010_d5c1_9f49_f77b5edb6fde_TB7896_11 ≡ -2 (mod 18)

Therefore, the answer is E, -2.

Evaluate the Function at Without Using a Calculator