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Number Theory SMC QuestionsAll the following questions were from the last 10 questions (i.e. Q16-25) of Senior Maths Challenge papers from 2005-2012).Positive integers x and y satisfy the equation x-11=y.What is the maximum possible value of xy?A 2 B 4 C 8 D 11 E 44If x2-px-q=0, where p and q are positive integers, which of the following could not equal x3?A 4x+3 B 8x+5 C 8x+7 D 10x+3 E 26x+5All the digits of a number are different, the first digit is not zero, and the sum of the digits is 36. There are N×7! such numbers. What is the value of N?A 72 B 97 C 104 D 107 E 128Four integers a, b, c and d are such that:abcd+abc+bcd+cda+dab+ab+bc+cd+da+ac+bd+a+b+c+d=2009What is the value of a+b+c+d?A 73 B 75 C 77 D 79 E 81Andy and his younger cousin Alice both have their birthdays today. Remarkably, Andy is now the same age as the sum of the digits of the year of his birth and the same is true of Alice. How many years older than Alice is Andy?A 10 B 12 C 14 D 16 E 18How many prime numbers p are there such that 199p+1 is a perfect square?A 0 B 1 C 2 D 4 E 8How many pairs of real numbers (x,y) satisfy the equation x+y2=x+3y-3?A 0 B 1 C 2 D 4 E infinitely manyThe length of the hypotenuse of a particular right-angled triangle is given by 1+3+5+7+…+25. The lengths of the other two sides are given by 1+3+5+…+x and 1+3+5+…+y where x and y are positive integers.What is the value of x+y?A 12 B 17 C 24 D 28 E 32Damien 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 some prime numbers. What is the smallest number of possible prime number divisors that Damien needs to check before he can be sure that 457 is a prime number?A 8 B 9 C 10 D 11 E 12The year 1789 (when the French Revolution started) has three and no more than three adjacent digits (7, 8 and 9) which are consecutive integers in increasing order. How many years between 1000 and 9999 have this property?A 130 B 142 C 151 D 169 E 180Which positive integer n satisfies the following equation:3n3+4n3+5n3+…+n3-5n3+n3-4n3+n3-3n3=60A 5 B 11 C 31 D 60 E 2006X is a positive integer in which each digit is 1; that is, X is of the form 11111...Give that every digit of the integer pX2+qX+r (where p, q and r are fixed integer coefficients and p>0) is also 1, irrespective of the number of digits X, which of the following is a possible value of q?A -2 B -1 C 0 D 1 E 2What is the sum of the values of n for which both n and n2-9n-1 are integers?A -8 B -4 C 0 D 4 E 8Given that S=x+20+x+21+x+22+…+x+100, where x is a positive integer, what is the smallest value of x such that S is a perfect square?A 1 B 2 C 4 D 8 E 64The factorial of n, written n!, is defined by n!=1×2×3×…×n-1×n. For how many positive integer values of k less than 50 is it impossible to find a value of n such that n! ends in exactly k zeros?A 0 B 5 C 8 D 9 E 10 ................
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