# Feb 2019 (CML)

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 Feb 24 – March 2.

Feb 24: What is the only number between 60 and 70 with an odd number of factors.

Answer: 64. If a number has an odd number of factors, then it has to be a square.

Feb 25: Find the sum of 2 prime numbers less than 100 that are both 1 more than a multiple of 5 and 2 less than a multiple of 3.

Answer: This number has unit digit 1. Among the prime numbers 11, 31, 41, 61 and 71, 31 and 61 are 2 less than a multiple of 3. So the two numbers are 31 and61, and their sum is 92.

Feb 26: Find the smallest perfect square number that satisfies the requirement: If it is decreased by one, then the new number is a multiple of 2, 3, 4, 5 and 6.

Answer: The multiples of 2, 3, 4, 5 and 6 are multiples of 60. The number 121 decreased by 1 becomes 120. So the answer is 121.

Feb 27: What is the remainder when $316^{2019}$ is divided by 10?

Feb 28: There are 4 ladies in the car. Their ages in all possible pairs are 96, 79, 84, 95, 100, 83. What is the sum of their ages?

Answer: a+b=96, b+c=79, c+d=84, a+c=95, a+d=100, b+d=83. If we add them all up, 3(a+b+c+d) = 96+79+84+95+100+83 = 537, a+b+c+d = 179.

March 1: If July 24 is a Sunday, then what day of the week will be Sept 24 of the same year?

Answer: July 31 is a Sunday, Aug 28 is a Sunday, Sept 4 is Sunday, Sept 24 is a Saturday.

March 2: The sum of the first 6 prime numbers is ___ more than the sum of the first six odd numbers. (Note 1 is not a prime number.)

Answer: The first 6 prime numbers are 2, 3, 5, 7, 11 and 13. The first six odd numbers are 1, 3, 5, 7, 9, 11. So the difference is 5.

Week of Feb 17 – Feb 23.

Feb 17: When Nick writes a composition, he always writes 11 words per line. The 1200th word in his composition will be the ___th word in the ___th line.

Answer: 1200/11 = 109 r 1. The 1200th word will be the 1st word in the 110th line.

Feb 18: If a number machine is fed an odd number, it multiplies it by 3 and then adds 1. If it is fed an even number, it divides it by 4. After the number 21 is fed into the machine, each result will be fed back into the machine. Before long, the only two numbers are produced by the machine are ___ and ___.

Answer: 21 -> 64 -> 16 -> 4 -> 1 -> 4 -> 1… The two numbers are 4 and 1.

Feb 19: How many numbers between 2999 and 4999 have the property that the sum of the digits is less than 6?

Answer: If the leading digit is 3, then the other 3 digits add up to at most 2. They can be 3000, 3001, 3010, 3100, 3002, 3020, 3200, 3011, 3101, 3110. If the leading digit is 4, then the other 3 digits add up to at most 1. They can be 4000, 4001, 4010, 4100. So there are 14 such numbers.

Feb 20: How many shares of stock must be purchased at $41 {1\over 6}$ dollars per share and sold at $41 {2\over 3}$ per share in order to make a profit of 100?

Feb 21: If $a^2+b^2+c^2=49$ where $a, b$ and $c$ are different positive integers. If $a and $a=2$, what is $c$?

Answer: $b^2+c^2 = 45$. The only option is $b=3$ and $latex $c=6$. Feb 22: If $y = {1\over {{1\over b}+{1\over a} }}$, find the value of$\latex y$when $a=3$ and $b=6$. Answer: $y = {1\over {1/3+1/6}} = 2$ Feb 23: Find the smallest 5-digit integer that is divisible by 2, 3, 4, 5, and 6. Answer: It ends with 0 (divisible by 5 and 2); the last two digits are divisible by 4 (divisible by 4); sum of the digits is divisible by 3 (divisible by 3). So the number is 10020. Week of Feb 10 – Feb 16. Click GoogleForm to submit answers by Sat, Feb 16. Feb 10: The digits 1357913579… are written on a screen. if 96 digits can be written on one line, then the first digit on the 5th line is ___. Answer: The first digit on the 5th line is 96*4+1=385th digit, which is 9. Feb 11: The first 4 odd numbers are 1, 3, 5, and 7. What is the 100th odd number? Answer: 199. Feb 12: 116 is divided into 3 parts proportional to 1, 1/4 and 1/3. The smallest of the three numbers is ___. Answer: $116 \cdot {{1\over 4} \over {1+{1\over 4}+{1\over 3}}} = {348 \over 19}$ Feb 13: What is the smallest integer $a$ so that $180a$ is a square number? Answer: 5 Feb 14: How many diagonals does a regular octagon have? Answer: 8 x 5 / 2 = 20. Feb 15: For what original price is 20% off the same as$30 off?

Answer: $150 Feb 16: Find the measure of the acute angle formed by the hour hand and minute hand on a clock at 12:15pm? Answer: The hour hand travels 30 degrees every hour. At 12:15pm, the hour hand is 7.5 degrees from 12:00pm since it has traveled 1/4 of an hour. The minute hand has traveled 90 degrees. Therefore, the angle between the two hands is 82.5 degrees. Week of Feb 3 – Feb 9. Happy Lunar New Year! Click GoogleForm to submit answers by Sat, Feb 9. Feb 3: An office building has 300 offices. Each office has a window or an AC unit or both. If 225 offices have a window and 200 offices have an AC unit, how many offices have both? Answer: 225+200-300 = 125 Feb 4: Mary spent$3 for plums, some at 30 cents a dozen and some at 40 cents a dozen. She sold all the plums at 3 cents each and made a profit of 24 cents. How many dozen plums did she buy at 40 cents a dozen?

Answer: Suppose Mary bought x dozen at 30 cents a dozen and y dozen at 40 cents a dozen. 30x+40y = 300, 6x-4y=24. Therefore x=6, y=3. Mary bought 3 dozen at 40 cents a dozen.

Feb 5: Mike counted all the ears, eyes, paws and tails on all the tigers in the zoo. The total he got is the same as the number of all the lions’ paws in the zoo. The total number of lions in the zoo must be a multiple of what integer?

Answer: 9T = 4L. The number of lions is a multiple of 9.

Feb 6: Mrs. E was born in 1974. When her daughter Linda was born, Mrs. E was 28 years old. In what year will Linda be 15 years old?