Number Theory Theory and questions for topic based enrichment activities/teaching Contents Prerequisites ................................................................................................................................................................................................................ 1 Questions ..................................................................................................................................................................................................................... 2 Investigative ............................................................................................................................................................................................................. 2 General Questions ................................................................................................................................................................................................... 2 Senior Maths Challenge Questions .......................................................................................................................................................................... 3 British Maths Olympiad Questions .......................................................................................................................................................................... 8 Key Theory ................................................................................................................................................................................................................... 9 Solutions..................................................................................................................................................................................................................... 13 Theory Questions ................................................................................................................................................................................................... 13 General Questions ................................................................................................................................................................................................. 15 SMC Questions ....................................................................................................................................................................................................... 20 British Maths Olympiad ......................................................................................................................................................................................... 33 Prerequisites For some questions, appreciation of the factorial function n! For one of the questions, knowledge of the difference between a convergent and divergent infinite series. For one of the questions, knowledge of logarithms and determining a turning point using calculus. Appreciation of big-sigma to indicate a summation over a range. Questions Investigative These questions are intended to highlight key theory in number theory or proofs you should be aware of (the latter of which you should research, as you would not be expected to prove them). For more general questions, see those in the next section or the SMC/BMO questions. 1. If x is divisible by a, then is x2 divisible by a? 2. If x2 is divisible by a, then is x divisible by a? More generally, if xn is divisible by a for some integer n, then is x divisible by a? 3. Prove that √2 is irrational. 4. Prove that is divergent. 5. Show that there are an infinite number of prime numbers. 6. Why is 1 not considered to be prime? General Questions 1. Show that the sequence 2, 5, 8, 11, 14, 17, … can never be a square number. 2. Show that 2n = n3 has no integer solution for n. 3. Does n have an integer solution to 4. Find integer solutions to xy + x + y = 2004 where x>0, y>0. 5. How many positive integer values of n is also an integer? 6. Suppose an integer has d distinct prime factors, where each is repeated ai times. How many divisors does the number have? How many factors does 1010 have? 7. Find all integer solution to the equation ab = ba that are non-trivial (ab), and show that these are the only solutions. 8. (a) Suppose n > 2 is a positive integer and that 2k is the highest power of 2 that divides any term in the sequence 2, 3, 4, ... , n. Show that only one term of the sequence is divisible by 2k. Deduce that lcm(2, 3, 4, ... , n) = 2kc where c is an odd integer. (b) By expressing the number in the form a/b, show that x cannot be an integer. 9. Show that if a number is divisible by 3, its digits add up to a multiple of 3. 10. Determine the largest power of 10 that is a factor of the following: (a) 50! (b) 1000! (c) In general, n! Senior Maths Challenge Questions Level 1 Level 3 1. How many prime numbers are there less than 20? 1. Which of these five expressions represents the largest A 6 B 7 C 8 number? 9 D 9 E 10 A 9(9 ) B 999 C 999 D (99)9 E 999 2. What is the remainder when the number 743589 × 301647 2. What is the value of is divided by 5? 1 + 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + A 0 B 1 C 2 2))))))))))? D 3 E 4 A 210 +1 B 211 −1 C 211 +1 D 212 −1 E 212 + 1 3. Exactly one of the following numbers is divisible by 11. 3. I am trying to do a rectangular jigsaw puzzle. The puzzle was Which is it? made by starting with a rectangular picture and then cutting 7 7 7 A 10 − 11 B 10 −1 C 10 it into 1000 pieces by sawing along the lines of a (wiggly!) 7 7 D 10 +1 E 10 + 11 rectangular grid. I start by separating out all the edge and corner pieces. Which of the following could not possibly be Level 2 the number of corner and edge pieces of such a jigsaw? 1. The mean of seven consecutive odd numbers is 21. What is A 126 B 136 C 216 the sum of the first, third, fifth and seventh of these D 316 E A-D are all possible numbers? A 16 B 21 C 84 4. Observe that 18 = 42 + 12 + 12 + 02. How many of the first D 147 E more information needed fifteen positive integers can be written as the sum of the squares of four integers? 2. The value of the product wxyz is 2002 and w, x, y and z are A 11 B 12 C 13 2 2 2 2 prime numbers. What is the value of w + x + y + z ? D 14 E 15 A 66 B 203 C 260 D 285 E 343 5. The probability of a single ticket winning the jackpot in the National Lottery is 6 5 4 3 2 1 A 2016 B 2025 C 2040 . 49 48 47 46 45 44 D 2048 E 2050 If I buy one ticket every week, approximately how often might I expect to win the jackpot? 10. Which of the following numbers n gives a counter-example A once every hundred years for the statement: ‘If n is a prime number then n2 + 2 is also B once every twenty thousand years a prime number’? C once every hundred thousand years A 3 B 5 C 6 D once every quarter of a million years D 9 E none of them E once every million years n 20 n 6. The factorial of n, written n!, is defined by n! = 1 × 2 × 11. When n = 81, what is the value of 3 ? 3×…×(n − 2) × (n − 1) × n 1 1 e.g. 6! = 1 × 2 × 3 × 4 × 5 × 6 = 720. What is the smallest A less than 100 B 3 positive integer which is not a factor of 50! ? C 1 D 3 E more than 100 A 51 B 52 C 53 D 54 E 55 12. The number of the year 2003 is prime. How many square numbers are factors of 20032003? 2 2 2 2 7. What is the value of (61 − 39 ) ÷ (51 − 49 )? A 0 B 1 C 44 A 10·5 B 11 C 12 D 1002 E 2003 D 21 E 22 13. A square number is divided by 6. Which of the following 8. Given that a and b are integers greater than zero, which of could not be the remainder? the following equations could be true? A 0 B 1 C 2 A a – b = a b B a + b = a b D 3 E 4 C a – b = a b D a + b = a – b E a+ b = a + b 9. Observe that 2000 = 24 × 53. What is the number of the next year after the year 2000 which can be written ab × cd where a, b, c, d are 2, 3, 4, 5 in some order? Level 4 6. What is the value of 22003 − 22002 − 22001 − 22000? A −22002 B 0 C 2−4000 1. What is the maximum possible value of the product of a list D 64 E 22000 of positive integers, which are not necessarily all different, given that the sum of these numbers is 100? 7. The value of 12004 + 32004 + 52004 + 72004 + 92004 is calculated 10 50 20 20 A 10 B 2 C 2 × 3 using a powerful computer. What is the units digit of the 20 2 32 D 5 E 2 × 3 correct answer? A 9 B 7 C 5 2. Sam correctly calculates the value of 58 × 85. How many D 3 E 1 digits does her answer contain? A 11 B 12 C 13 D 14 E 15 8. Which is the lowest positive integer by which 396 must be multiplied to make a perfect cube? 3. Note that 2001 = 3 × 23 × 29. What is the number of the A 11 B 66 C 99 next year which can be written as the product of three D 121 E 726 distinct primes? A 2002 B 2004 C 2006 9. How many two-digit numbers N have the property that the D 2007 E none of these sum of N and the number formed by reversing the digits of N is a square? 4. For how many integer values of n does the equation x2 + nx A 2 B 5 C 6 − 16 = 0 have integer solutions? D 7 E 8 A 2 B 3 C 4 D 5 E 6 10. Damien wishes to find out if 457 is a prime number. In order to do this he needs to check whether it is exactly divisible by 5. Which of the following is divisible by 3 for every whole some prime numbers. What is the smallest number of number x? possible prime number divisors that Damien needs to check A x3 − x B x3 − 1 C x3 before he can be sure that 457 is a prime number? D x3 + 1 E x3 + x A 8 B 9 C 10 D 11 E 12 Level 5 6. Let N be a positive integer less than 102002. When the digit 1 is placed after the last digit of N, the number formed is 1. Three people each think of a number which is the product three times the number formed when the digit 1 is placed in of two different primes. Which of the following could be the front of the first digit of N. How many different values of N product of the three numbers which are thought of? are there? A 120 B 144 C 240 A 1 B 42 C 333 D 3000 E 12100 D 667 E 2002 2.