Exam 11: Number Theory

Put 0 and 2 (one of each) into the blanks to make a true statement. Explain. If it is not possible, explain why. _____ is a multiple of _____ because _____.

Give the prime factorization of n, where n = 4 × 7²⁰ × 5000. If it is not possible, explain why not.

Give the prime factorization of n, where n = 13⁷ × 3000.

As a charitable service, your class undertakes a project where they fill backpacks with donated school supplies for underprivileged children. The donations include 135 notebooks, 216 pencils, and 81 pens. You want to use all the donations and include the same number of each item in each backpack. What is the largest number of backpacks you can fill, and how many items will be in each backpack?

Suppose 7 is not a factor of n. Can 21 be a factor of n? If 21 can be a factor of n, give an example for n. If 21 cannot be a factor of n, give an explanation from basic principles.

Hamburger patties come in packages of 16, and hamburger buns come in bags of 12. How many of each do you need to buy so that you have the same number of buns as you do hamburgers?

Use the word multiple to say that 360 is a factor of N.

Note: This is a knowledge of number theory question. Do not use a calculator for this question. Is it possible for some choice of positive whole numbers m and n that 75^m = 25ⁿ? Justify your decision.

Note: This is a theory question. Do not use a calculator for this question. Is it possible for some choice of positive whole numbers m and n, such that 45^m = 15ⁿ? Justify your answer.

For each part, give an example, if one exists. If there is no example, explain why not. A) A whole number that has 15, 21, and 1000 as factors but does not have 9 as a factor B) A prime number that has 7 and 19 as factors and is not a perfect square

Two neighboring satellites send out signals at regular intervals. One sends a signal every 180 seconds, and the other sends a signal every 280 seconds. If both satellites send out a signal at 12:00 midnight on January 1, when will be the next time that they both send out a signal at the same time?

Note: This is a knowledge of number theory question. Do not use a calculator for this question. A) Select any choice that is a factor of 62,296,715,880 that is equal to 2³ × 3² × 5 × 7 × 47² × 31 × 19². 15 16 21 75 94 217 n = 19 × 31² B) Explain how you know that your answers in A are correct, even without calculation.

In each part, find a whole number for m to make the equality true. If it is not possible, explain why. For credit, your work should show an understanding of number theory. A) 5² × 10³ × 17⁶ = 2³ × 17⁶ × m B) 5² × 7⁶ × 11⁴ = 5 × 35⁶ × 11⁴ × m

If we write the first 10,000 numbers in six columns, as started below, then 9999 would be in the fifth column. Write enough (numbers, words) to make your thinking clear. 1 2 3 4 5 6 7 8 9 10 11 12

Suppose K = $2 ^ { 5 } \cdot 7 \cdot 11$ , L = $2 ^ { 3 } \cdot 7 \cdot 11 \cdot 13$ , M = $2 \cdot 29 ^ { 2 }$ , and N = $4 \cdot 11 \cdot 13 ^ { 2 } \cdot 29$ . Name the greatest common factor of each of the following (in factored form). A) K and N B) K and L C) M and N D) K, L, and M

Every two different prime numbers are relatively prime.

There are no values of b and c for which 27^b = 9^c. Explain your choice.

Tell the difference between (a) "give a prime factor of 350" versus "give a prime factorization of 350," and (b) "give a number that has an odd factor" versus "give a number that has an odd number of factors."

A band has been invited to march at the Rose Parade and needs to make money to cover the expenses. They divide up into three teams and shovel snow from long driveways for four days before Christmas. The first team makes $315, the second $240, and the third $210. If they charged the same whole-dollar rate for each driveway, what was that rate? Explain.

If 27 is a factor of n, then n is a multiple of 27.