Sept 2017 (AMC 8)

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.

This week we continue to focus on numbers and number theory, which deals with integers.

Click GoogleForm to submit answers by Sat, Sept 30.

Week of Sept 24 – 30

Sept 24: Each of the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 is used only once to make two five-digit numbers so that they have the largest possible sum. Which of the following could be one of the numbers? $\textbf{(A)}\hspace{.05in}76531\qquad\textbf{(B)}\hspace{.05in}86724\qquad\textbf{(C)}\hspace{.05in}87431\qquad\textbf{(D)}\hspace{.05in}96240\qquad\textbf{(E)}\hspace{.05in}97403$

Answers: C.  The ten-thousands digit must be 9 or 8, the thousands digit must be 7 or 6, etc.

Sept 25: Bag A has three chips labeled 1, 3, and 5. Bag B has three chips labeled 2, 4, and 6. If one chip is drawn from each bag, how many different values are possible for the sum of the two numbers on the chips? $\textbf{(A) }4 \qquad\textbf{(B) }5 \qquad\textbf{(C) }6 \qquad\textbf{(D) }7 \qquad\textbf{(E) }9$

Answer: B. The sum can be 3, 5, 7, 9 or 11.

Sept 26: How many digits are in the product $4^5 \cdot 5^{10}$? $\textbf{(A) } 8 \qquad\textbf{(B) } 9 \qquad\textbf{(C) } 10 \qquad\textbf{(D) } 11 \qquad\textbf{(E) } 12$

Answer: D. $2^{10}\cdot 5^{10}= 10^{10}$.

Sept 27: Let $w$ $x$ $y$, and $z$ be whole numbers. If $2^w \cdot 3^x \cdot 5^y \cdot 7^z = 588$, then what does $2w + 3x + 5y + 7z$ equal? $\textbf{(A) } 21\qquad\textbf{(B) }25\qquad\textbf{(C) }27\qquad\textbf{(D) }35\qquad\textbf{(E) }56$

Answer: A.  w=2, x=1, y=0, z=2.

Sept 28: What is the tens digit of $7^{2011}$? $\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }3\qquad\textbf{(D) }4\qquad\textbf{(E) }7$

Answer: D. The last 2 digits repeat in a cycle of length 4: 07, 49, 43, 01, 07, 49, 43, 01…

Sept 29: How many 4-digit positive integers have four different digits, where the leading digit is not zero, the integer is a multiple of 5, and 5 is the largest digit? $\textbf{(A) }24\qquad\textbf{(B) }48\qquad\textbf{(C) }60\qquad\textbf{(D) }84\qquad\textbf{(E) }108$

Answer D: If the unit digit is 5, the thousands digit has 4 options (1,2,3,4), the hundreds digit has 4 options and tens has 3 options. If the unit digit is 0, then 5 can appear in tens, hundreds or thousands digit. The remaining 2 digits have 4*3 options.  So in total 4*4*3 + 3*4*3.

Sept 30: In how many ways can 10001 be written as the sum of two primes? $\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }3\qquad\textbf{(E) }4$

Answer: A.  One of the two primes must be 2 since 2 odd primes add to an even sum. However, 2+9999=10001 and 9999 is not prime.

This week we focus on numbers and number theory, which deals with integers.

Click GoogleForm to submit answers by Sat, Sept 23.  Last week’s problems are also due Sept 23.

Week of Sept 17 – 23

Sept 17: If $3^p + 3^4 = 90$ $2^r + 44 = 76$, and $5^3 + 6^s = 1421$, what is the product of $p$ $r$, and $s$? $\textbf{(A)}\ 27 \qquad \textbf{(B)}\ 40 \qquad \textbf{(C)}\ 50 \qquad \textbf{(D)}\ 70 \qquad \textbf{(E)}\ 90$

Answer: B.  p=2, r=5, s=4

Sept 18: The sum of six consecutive positive integers is 2013. What is the largest of these six integers? $\textbf{(A)}\ 335 \qquad \textbf{(B)}\ 338 \qquad \textbf{(C)}\ 340 \qquad \textbf{(D)}\ 345 \qquad \textbf{(E)}\ 350$

Answer: B.  2013/3 = 671.   333, 334, 335, 336, 337, 338

Sept 19: How many 4-digit numbers greater than 1000 are there that use the four digits of 2012? $\textbf{(A)}\hspace{.05in}6\qquad\textbf{(B)}\hspace{.05in}7\qquad\textbf{(C)}\hspace{.05in}8\qquad\textbf{(D)}\hspace{.05in}9\qquad\textbf{(E)}\hspace{.05in}12$

Answer: D. 6 numbers begin with 2, 3 numbers begin with 1.

Sept 20: What is the ratio of the least common multiple of 180 and 594 to the greatest common factor of 180 and 594? $\textbf{(A)}\ 110 \qquad \textbf{(B)}\ 165 \qquad \textbf{(C)}\ 330 \qquad \textbf{(D)}\ 625 \qquad \textbf{(E)}\ 660$

Answer: C.  Prime factorization.

Sept 21: When Clara totaled her scores, she inadvertently reversed the units digit and the tens digit of one score. By which of the following might her incorrect sum have differed from the correct one? $\textbf{(A)}\ 45 \qquad \textbf{(B)}\ 46 \qquad \textbf{(C)}\ 47 \qquad \textbf{(D)}\ 48 \qquad \textbf{(E)}\ 49$

Answer: A. The difference has to be a multiple of 9.

Sept 22: The smallest number greater than 2 that leaves a remainder of 2 when divided by 3, 4, 5, or 6 lies between what numbers? $\textbf{(A)}\hspace{.05in}40\text{ and }50\qquad\textbf{(B)}\hspace{.05in}51\text{ and }55\qquad\textbf{(C)}\hspace{.05in}56\text{ and }60\qquad\textbf{(D)}\hspace{.05in}\text{61 and 65}\qquad\textbf{(E)}\hspace{.05in}\text{66 and 99}$

Answer: least common multiple of 3, 4, 5, and 6 is 60.  60+2=62.

Sept 23: What is the smallest positive integer that is neither prime nor square and that has no prime factor less than 50? $\textbf{(A)}\hspace{.05in}3127\qquad\textbf{(B)}\hspace{.05in}3133\qquad\textbf{(C)}\hspace{.05in}3137\qquad\textbf{(D)}\hspace{.05in}3139\qquad\textbf{(E)}\hspace{.05in}3149$

This is the first week for the 2017-2018 season of SMC.   As in the past 2 years, I’ll post problems on a common topic each week.  This week, we will focus on calculating prices, discounts and change.  For this week, since we are restarting, you have 2 weeks to finish the problems. Please submit your answers to me by Saturday, Sept 23.  (Usually you have 1 week.)

Week of Sept 10 – 16

Sept 10: Margie bought $3$ apples at a cost of $50$ cents per apple. She paid with a 5-dollar bill. How much change did Margie recieve? $\textbf{(A) }\ \textdollar 1.50 \qquad \textbf{(B) }\ \textdollar 2.00 \qquad \textbf{(C) }\ \textdollar 2.50 \qquad \textbf{(D) }\ \textdollar 3.00 \qquad \textbf{(E) }\ \textdollar 3.50$

Sept 11: A sign at the fish market says, “50% off, today only: half-pound packages for just $3 per package.” What is the regular price for a full pound of fish, in dollars? $\textbf{(A)}\ 6 \qquad \textbf{(B)}\ 9 \qquad \textbf{(C)}\ 10 \qquad \textbf{(D)}\ 12 \qquad \textbf{(E)}\ 15$ Answer: D Sept 12: Eight friends ate at a restaurant and agreed to share the bill equally. Because Judi forgot her money, each of her seven friends paid an extra$2.50 to cover her portion of the total bill. What was the total bill? $\textbf{(A)}\ \text{\textdollar}120\qquad\textbf{(B)}\ \text{\textdollar}128\qquad\textbf{(C)}\ \text{\textdollar}140\qquad\textbf{(D)}\ \text{\textdollar}144\qquad\textbf{(E)}\ \text{\textdollar}160$

Sept 13: At the 2013 Winnebago County Fair a vendor is offering a “fair special” on sandals. If you buy one pair of sandals at the regular price of $50, you get a second pair at a 40% discount, and a third pair at half the regular price. Javier took advantage of the “fair special” to buy three pairs of sandals. What percentage of the$150 regular price did he save? $\textbf{(A)}\ 25\% \qquad \textbf{(B)}\ 30\% \qquad \textbf{(C)}\ 33\% \qquad \textbf{(D)}\ 40\% \qquad \textbf{(E)}\ 45\%$

Sept 14: A shop advertises everything is “half price in today’s sale.” In addition, a coupon gives a 20% discount on sale prices. Using the coupon, the price today represents what percentage off the original price? $\textbf{(A)}\hspace{.05in}10\qquad\textbf{(B)}\hspace{.05in}33\qquad\textbf{(C)}\hspace{.05in}40\qquad\textbf{(D)}\hspace{.05in}60\qquad\textbf{(E)}\hspace{.05in}70$

Sept 15: Jamar bought some pencils costing more than a penny each at the school bookstore and paid $\textdollar 1.43$. Sharona bought some of the same pencils and paid $\textdollar 1.87$. How many more pencils did Sharona buy than Jamar? $\textbf{(A)}\hspace{.05in}2\qquad\textbf{(B)}\hspace{.05in}3\qquad\textbf{(C)}\hspace{.05in}4\qquad\textbf{(D)}\hspace{.05in}5\qquad\textbf{(E)}\hspace{.05in}6$
Setp 16: The taxi fare in Gotham City is $2.40 for the first $\frac12$ mile and additional mileage charged at the rate$0.20 for each additional 0.1 mile. You plan to give the driver a $2 tip. How many miles can you ride for$10? $\textbf{(A) } 3.0\qquad\textbf{(B) }3.25\qquad\textbf{(C) }3.3\qquad\textbf{(D) }3.5\qquad\textbf{(E) }3.75$