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Number TheoryQuestion and Answers: Page 12
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Question and Answers Forum |
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Number TheoryQuestion and Answers: Page 12 |
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| _≤^≥ = SOLVE the EQUATION_ ^ _( •) n−⌊(√n)⌋−⌊(n)^(1/3) ⌋+⌊(n)^(1/6) ⌋=2016 |
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| Find ∣{n∈N∣n≤100, 4!∣2^n −n^3 }∣. (Note that ∣S∣ denotes the cardinality or number of elements of a set,S). |
| (1/(5!!)) + ((1.3)/(7!!)) + ((1.3.5)/(9!!)) + ... + ((1.3.5.7....95)/(99!!)) =? |
| Suppose x,y,z ∈N, (yz+x) is prime (yz+x)∣(zx+y), (yz+x)∣(xy+z). Find all possible values of (((xy+z)(zx+y))/((yz+x)^2 )). |
| Let N be the greatest multiple of 36 all of whose digits are even and no two of whose digits are the same. Find the remainder when N is divided by 1000. |
| Let K be the product of all factors (b−a) (not necessarily distinct) where a and b are integers satisfying 1≤a≤b≤10. Find the greatest integer n such that 2^n divides K. |
| How many triples of positive integers (x,y,z) satisfy 79x+80y+81z =2016 |
| Assuming FLT, prove Fermat−Euler theorem: (a,n) =1,n≥2⇒a^(∅(n)) ≡1(mod n) |
| How many natural numbers less than 1000 have the sum of their digits equal to 5? |
| What is the minimum value obtained when an arbitrary number of three different non−zero digits is divided by the sum of its digits? |
| Find four values of n satisfying 1≤n≤2000 and 2^n =n^2 (mod 1024) |
| z is a complex number with Re(z) , Im(z)∈N. Determine z if z.z^− =1000 |
| Find the number of rational numbers r, 0<r<1, such that when r is written as a fraction in lowest term. The numerator and the denominator have a sum of 1000. |
| GCD of two unequal numbers can′t exceed their absolute difference. Prove. |
| How many ways can 2018 be expressed as the sum of two squares? |
| Find the number of rational numbers r, 0<r<1, such that when r is written as fraction in lowest term. The numerator and demominator have a sum of 1000. |
| Let a,b and c be positive integers such that ab+1∣bc+1 and bc+1∣ca+1. Show that ab+1 is the sum of two squares. |
| Evaluate 5!•6!(mod 7!) |
| Find the sum of all positive two−digit integers that are divisible by each of their digits. |
| Let x and y be integers such that xy≠1, x^2 ≠y and y^2 ≠x. (i) Show that p∣xy−1 and p∣x^2 −y then p∣y^2 −x where p is a prime. (ii) Let p be a prime. Suppose that p∣x^2 −y and p∣y^2 −x, must p∣xy−1? [If yes, then prove it. If no, then give a counter example] |
| Let n∈N. Using the formula lcm(a,b) = ((ab)/(gcd(a,b))) and lcm(a,b,c) =lcm(lcm(a,b),c), find all the possible value of ((6•lcm(n,n+1,n+2,n+3))/(n(n+1)(n+2)(n+3))) |
| Let a,b and c be positive integers such that ab+1∣bc+1 and bc+1∣ca+1. Show that ab+1 is the sum of two squares. |
| The Diophantine equation x^2 +y^2 +1 =N(xy+1) has infinitely many integer solutions if N equals? Any help please? |
| 17x ≡ 3 (mod 29) |