Number Sense/Theory

Prepared by Charles Linneman

Jewel Middle School

1501 Waterford Road

North Aurora, IL 60542

(clinneman@sd129.org)

  1. What is the smallest positive integer which gives a remainder of :

  1. 1 when divided by 4, and 4 when divided by 7?
  2. 1 when divided by 4, 4 when divided by 7, and 6 when divided by 9?
  3. 1 when divided by 2, 1 when divided by 3, and 2 when divided by 7?
  4. 2 when divided by 3, 3 when divided by 4, and 11 when divided by 13; which is also greater than 13?
  5. 1 when divided by 4, 4 when divided by 7, and 6 when divided by 11?
  1. When the three integers 347, 473, and 788 are divided by a positive integer, d, where

    2. d > 1, the remainders are the same. What is the smallest possible value of d? What is the largest possible value of d? How many possible values of d are there?

  2. What is the greatest common factor (GCF) of 738, 774, and 873?
  3. What is the remainder when 7 to the 2001 power (7^2001) is divided by 13?
  4. What is the units digit in the product of 667^1 x 667^2 x 667^3 x 667^4 x ….. x 667^1999 x 667^2000 x 667^2001 where first 2001 natural number powers of 667 are multiplied?
  5. Mr. Linneman’s favorite Easter candy is packaged in containers of 7 pieces or 12 pieces. What is the largest finite number of pieces he cannot obtain by only getting full containers of the candy?
  6. What is the number of positive integer factors of 840,840?
  7. What is the third smallest positive integer with exactly 5 positive integer divisors?
  8. How many positive integers less than or equal to 150 have exactly 12 positive integer divisors?
  9. What is the probability that a randomly chosen factor of 6^90 is a multiple of 48^13? Express your answer in fraction form.
  10. What is the largest integer value of n for which 9^n evenly divides 200!
  11. How many zeros does 300! end in?
  12. How many natural number perfect squares less than or equal to 900 are also perfect cubes?
  13. How many different 5-digit numbers can be formed using the digits 0,3,4,5,6,and 7 such that no digits repeat and the number is divisible by four?
  14. When creating a 5 digit number which is divisible by 36, by filling in the units and hundreds digit in 13,_6_ , what is the positive difference between the largest and smallest number which can be formed?
  15. Farmer Chuck bought 88 bales of hay for his goats. He wrote out a check for the purchase and also wrote the amount of the check on the palm of his hand, in order to enter the amount into his checkbook ledger when he arrived home. However on the trip home two of the digits were smeared out from the steeling wheel. All he could read was the middle 3 digits of $_96.3_. All farmer Chuck could remember was that each bale cost the same amount, was less than $10, and was the current market value rounded to a whole number of cents. What was the price farmer Chuck paid for each bale of hay?
  16. One-fourth of the boys left school early to play in the championship football game, leaving the ratio of boys to girls at 3:4. Later in the day 49 girls, who were either on the cheerleader or pom-pom squad, left school early on a bus to the game. This now made the ratio of girls to boys at still in school at 3:4. How many boys left school early to play in the championship football game?
  17. A pouch only contains red and yellow marbles. If 5 yellow marbles are removed from the pouch, the probability of drawing a yellow marble from the remaining marbles would be 3/7. If, instead, 9 more red marbles are added to the pouch, the probability of drawing a red marble would then be 7/12. If no changes are actually made, what is the positive difference between the number of red marbles and yellow marbles in the pouch?
  18. Natalie is going to mail a package using the exact postage. She will either use all 21 cent or 33 cent stamps. She would need 8 more 21 cent stamps than 33 cent stamps. How many dollars to the nearest cent will it cost Natalie to mail her package?
  19. All together Rita has 25 coins worth $4.15 consisting of only quarters and dimes in her car for paying highway tolls. How many of the coins are dimes?
  20. Marshall first five test scores were 92, 80, 89, 74, and 78. What is the lowest score Marshall can get on his next test in order to have an average of at least 85?
  21. The average of Matthew’s school, chapter, and state competition scores was a 41. What does Matthew need to get for his national competition score in order to raise his average to a 42?
  22. After math team practice Matt loves to play his favorite video game, Asteroid Blaster. The game saves and displays Matt’s five highest game scores. His highest score is 3783 and his lowest of the five high scores is 3467. The average of his five high scores is also displayed and is 3634. The next time Matt plays the game he gets a new high score of 3887. What will be the new displayed average of his five highest scores?
  23. A set of 7 whole numbers has a median of 37, mode of 39, and a mean of 35. What is the positive difference between the greatest and the least possible ranges of the set of numbers?