Exam 11: Number Theory

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 least common multiple of each of the following (in factored form). A) K and L B) M and N C) K and M D) K, L, and N

A) What is the least common multiple of 1485 and 792 (in factored form)? B) Write two other common multiples of 1485 and 792.

Is 245 a prime number? Explain.

Five is a multiple of zero. Explain.

Explain, without extensive calculation, why the following equation can or cannot be correct. 17² × 19² × 37⁵ = 18⁴ × 41⁴ (In your explanation, the grader will look for a clear reference to a major theoretical result.)

Determine if the statements are true or false. Explain your choice. A) Every whole number is a multiple of itself. B) It is possible for an even number to have an odd factor. C) Zero is a multiple of every whole number. D) 2⁵⁰ is a factor of 100³⁰.

Use the word factor to say that M is a multiple of 240.

A) Use the prime factorizations of 345, 264, and 495 to find the least common multiple of the three numbers. B) Compute the following: $\frac { 345 } { 495 } + \frac { 250 } { 264 }$ . (Leave the answer in factored form.)

Note: This is a knowledge of number theory question. Do not use a calculator for this question. Without calculation, explain why Romeo and Juliet can or cannot both be correct when they are talking about the same large number: Romeo: "The number is 7 × 11 × 17² × 37 × 67 × 97." Juliet: "The number is 3 × 11 × 21² × 37 × 67 × 89."

Determine whether m and n are primes. Write only enough to make your decisions clear. A) m = 23 × 29 (= 667) B) n = 133

What, if anything, can you say about the oddness or evenness of m: A) when 5,063,338 × m is an even number? B) when 5,063,338 + m is an even number?

What is the LARGEST prime number that you need to test in checking for the primeness of the following? Explain your choice. A) 173 B) 982

If it is possible, give a whole number that is relatively prime to 24. If it is not possible, explain why.

Does zero have any factors? Explain your answer.

Fill in the blanks to make a true sentence or state if no number or algebraic expression will make the sentence true. A) An example of a number that has an odd number of factors is _____. B) If n = 13^{8 .} 17¹⁰, then the prime factorization of 26 ^. n is _____. C) Three will be a factor of 1,400,000,00?,000,000,014 if the missing digit (?) is _____ or _____ or _____.

Determine if the following statements are true or false. A) Every whole number is a factor of itself. B) It is possible for an odd number to have an even factor. C) Zero is a factor of every whole number. D) 5²⁰ is a factor of 50¹².

There are no values of r and s for which 11^r = 9^s. Explain your choice.