1) True or False? If false, then correct with a similar
statement. (In this
problem, as sume that variables stand for natural numbers.)
(a) If n is even, then n/2 is even.
(b) If n is even, then 4|2n.
(c) If n is even, then 4|n2.
(d) The general form for a multiple of three is 2a + 1.
(e) If 2|a and 5|b, then a2b is a multiple of 50.
(f) The sum of two oddnumbers is never a multiple of three.
(g) If 3|a and 7|b, then 10|a + b.
(h) If 3|a and 9|b, then 3|a + b.
(i) The sum of any three consecutive numbers is divisible by three .
(j) The product of any three consecutive numbers is divisible by six.
(k) If c|ab, then c|a or c|b.
(l) If d|a, then d2|a3.
(m) The square of any odd number leaves remainder 1 when you divide it
(n) The least common multiple of a and b always divides ab.
(2) Is 101 prime? Explain how to check this efficiently.
(a) Multiples of 2, Multiples of 3, Multiples of 6
(b) Multiples of 8, Multiples of 6, Multiples of 5
(c) Multiples of 4, Multiples of 16, Multiples of 64
(d) Multiples of 15, Multiples of 25
(What is a description of the intersection?)
(12) The gcd of two numbers is 18. Their lcm is 630. What are the two numbers?
(13) * Suppose there is a row of one million pennies, all with the heads side
facing up. Now someone goes through and flips over every second coin
(that is, the ones numbered 2, 4, 6, and so on are flipped to the tails side).
Now someone goes through and flips over every third coin (so they turn
over 3, 6, 9, and so on: now 3 is tails up but 6 is turned back to heads
up). This process is continued for every divisor from two to a million. Will
the coin numbered 2008 be heads up or tails up at the end of the process?
Which coins are heads up?
(14) * Find five consecutive composite numbers over 1000. Give a method for
finding ten consecutive composite numbers.
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