# Sept 2018 (AMC8-MC)

I will post seven new problems every weekend, one for each day of the week. The idea is that you work on your own during the week and I will post answers the following Saturday.  For those of you who practiced with me last year, the process is similar.

For instructions on submitting weekly answers, click on the “Logistics” tab.
For previously posted problems, click on the “Archive” tab.
For previous prizes, click on the “News” tab.

Week of Sept 23 – Sept 29: All to do with exponents. How many of the following can you do in your head?

Sept 23: Express $2^5 \cdot 8^3 \cdot 16^2$ as a power of 4.

Answer: $4^{11}$

Sept 24: Express $2^2 \cdot 4^2 \cdot 8^2 \cdot 16^2 \cdot ... \cdot 1024^2$ as a power of 2.

Answer: $2^{110}$

Sept 25: Express $3^{16}$ as a power of $1\over 9$.

Answer: ${1\over 9}^{-8}$

Sept 26: When the expression $8^{10} \cdot 5^{22}$ is multiplied out, how many digits does the number have?

Answer: $2^8\cdot 10^{22} = 256\cdot 10^{22}$ has 25 digits

Sept 27: What is the sum of the digits of the number $2^{2018} \cdot 5^{2020} \cdot 3$?

Answer: $10^{2018}\cdot 75$. The sum of digits is 12.

Sept 28: What is the positive integer $N$ for which $22^2 \cdot 55^2 = 10^2\cdot N^2$?

Answer: $N = 121$

Sept 29: What is the value of $n$ such that $n\cdot 3^4 \cdot 2^5 = 6^6$?

Week of Sept 16 – Sept 22: Can you do the following calculations in your head?

Sept 16: What is the product of $38 \cdot 42$?

Answer: $(40-2)\cdot (40+2) = 40^2-2^2 = 1596$

Sept 17: What is the sum of the first 40 positive integers?

Answer: $41\cdot 20 = 820$

Sept 18: What is the value of 123,123 divided by 1001?

Sept 19: What is the sum (-100) + (-99) + (-98) + … + 97 + 98?

Sept 20: What is sum (-39) + (-37) + (-35) + … (-3) + (-1)?

Answer: $-20 \cdot 20 = -400$

Sept 21: What is the value of 100 – 99 + 98 – 97 + … + 4 – 3 + 2 – 1?