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Question: need help with all questions please 1 2 3 4...

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NEED HELP WITH ALL QUESTIONS PLEASE

1)

(1 point) Find the least integer n such that f(x) is O(for each of the following functions: (a) f(x) 4 log(a) (b) f(x) 3x5 + x4 (c) f(x)21 r2+1

2)

Enter T or F depending on whether the statement is true or not. (You must enter T or F - - True and False will not work.) If it is true do a proof and if it is false provide a counter-example (Of course the proofs and counter-examples must be written down on paper!) EEE 1. A necessary condition for an integer to be divisible by 8 is that it is divisible by 2. i 2. The square root of any natural number n that is even is even. 3. The product of 2 even numbers is even. li4. The product of 2 consecutives integers is even. EE 5.There exists an integer n such that 2-1 n EEE 6. If a divides b then a divides b2

3)

Enter Tor F depending on whether the statement is a true proposition or not. (You must enter T or F True and False will not work.) 1. 12 | 169 2. 111 is divisible by 11 3. 27214 is divisible by 193 4. 10 is divisible by 170 5. 19 228 6. 168 divides 12 Ei:7. 156 is divisible by 13 8. 110 | 10 9.34409 is divisible by 187

4)

(1 point) Suppose an integer x is divisible by 84 What other positive integers must also divide r? Hint: Think about the transitivity of divides. Enter your answer as a comma-separated list.] 1, 2, 3, 4, 6, 7, 12,14, Suppose an integer a is divisible by 24. What other positive integers must also divide a? [Enter your answer as a comma-separated list.] 1,2,3,4,6,8,12

5)

(1 point) For each of the following numbers write down the numbers prime factors as a comma-separated list (so, 5 or 2,3 for 25 and 12 respectively, but without the quotes) The prime factors of 60 are The prime factors of 50 are The prime factors of 20 are The prime factors of 84 are The prime factors of 95 are The prime factors of 46 are The prime factors of 31 are The prime factors of 91 are The prime factors of 26 are

6)

(1 point) What is the smallest integer, larger than 1, which is both a perfect square, and a perfect cube answer What is the smallest integer with prime factorization pq, where p and q are distinct prime numbers? answer: What are the next three smallest integers with prime factorization pq with p and q distinct? Smallest Largest What is the smallest integer with prime factorization pqr, where p, q, and r are distinct prime numbers? answer.

7)

8)

(1 point) Find the prime factorization of the follwing numbers (write p if a prime does not appear in the given number.) 90 -2a35c7111e13 17919h where i b- 2 i d- 0 0 l h- 0 280- 2a35c7d11e13 17919h where 12716 2 3 5c 7d11 13 17919h where с 0 0 2 41327 2a 3b 5c7d11 13 17919h where i d

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