Lesson Solving equations in integer numbers
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<H2>Solving equations in integer numbers</H2> <H3>Problem 1</H3>Solve equation in integer numbers {{{2^(x+1)}}} = {{{3^(y+2)}}} - {{{3^y}}}. <B>Solution</B> <pre> Left side of the given equation is {{{2^(x+1)}}}. Right side of the given equation is {{{9*3^y}}} - {{{3^y}}} = {{{8*3^y}}}. So, the given equation is equivalent to {{{2^(x+1)}}} = {{{8*3^y}}}. Due to uniqueness of decomposition integer numbers into the product of prime numbers, from the last equation we conclude x + 1 = 3, y = 0. <U>ANSWER</U>. x = 2; y = 0. </pre> <H3>Problem 2</H3>Solve equation in integer numbers {{{2^(x+1)}}} + {{{2^x}}} = {{{3^(y+2)}}} - {{{3^y}}}. <B>Solution</B> <pre> Left side of the given equation is {{{2*2^x}}} + {{{2^x}}} = {{{3*2^x}}}. Right side of the given equation is {{{9*3^y}}} + {{{3^y}}} = {{{8*3^y}}}. So, the given equation is equivalent to {{{3*2^x}}} = {{{2^3*3^y}}}. Due to uniqueness of decomposition integer numbers into the product of prime numbers, from the last equation we conclude x = 3, y = 1. <U>ANSWER</U>. x = 3; y = 1. </pre> <H3>Problem 3</H3>Find the solution in positive integer numbers x, y to equation {{{3x^4 - x^3*y-9317 = 0}}}. <B>Solution</B> <pre> Given equation is equivalent to 3x^4 - x^3*y = 9317, or x^3*(3x - y) = 9317. Integer number 9317 has the primary decomposition 9317 = 7*11^3. From uniqueness of prime decomposition for integer numbers, we conclude that EITHER x = 1, 3x-y = 9317 OR x = 11 and 3*11 - y = 7; y = 33-7 = 26. First option produces negative value of y; therefore, we discard it. The second option gives the <U>ANSWER</U> to the problem x= 11, y = 26. </pre> <H3>Problem 4</H3>Solve equation {{{12^x*24^y}}} = {{{216}}} in integer numbers. <B>Solution</B> <pre> Left side is {{{12^x*24^y}}} = {{{2^(2x+3y)*3^(x+y)}}}. Right side is 216 = {{{6^3}}} = {{{2^3*3^3}}}. So, equation is {{{2^(2x+3y)*3^(x+y)}}} = {{{2^3*3^3}}}. Due to uniqueness of the decomposition of integer numbers into the product of primes, it implies 2x + 3y = 3, (1) x + y = 3. (2) So, we have a system of two equations. It can be easily solved by Substitution or by Elimination. Let's do it using substitution x = 3 - y from equation (2). Then equation (1) will give you 2(3-y) + 3y = 3 6 - 2y + 3y = 3 y = 3 - 6 = -3. Then from (2), x = 3 - y = 3 - (-3) = 6. <U>ANSWER</U>. x = 6, y = -3. You can make check on your own to convince yourself that the answer is correct. // I did such check for myself. </pre> My other lessons in this site on miscellaneous problems on divisibility of integer numbers are - <A HREF=https://www.algebra.com/algebra/homework/divisibility/lessons/Light-flashes-on-a-Christmas-tree-and--a-Least-Common-Multiple.lesson>Light flashes on a Christmas tree and a Least Common Multiple</A> - <A HREF=http://www.algebra.com/algebra/homework/divisibility/lessons/The-number-that-leaves-a-remainder-1-when-divided-by-2-by-3-by-4-by-5-and-so-on-until-9.lesson>The number that leaves a remainder 1 when divided by 2, by 3, by 4, by 5 and so on until 9</A> - <A HREF=https://www.algebra.com/algebra/homework/divisibility/The-number-rem4-mod7-rem5-mod8-and-rem6-mod9.lesson>The number which gives remainder 4 when divided by 7, remainder 5 when divided by 8 and remainder 6 when divided by 9</A> - <A HREF=https://www.algebra.com/algebra/homework/divisibility/Introductory-problems-on-divisibility-numbers.lesson>Introductory problems on divisibility of integer numbers</A> - <A HREF=https://www.algebra.com/algebra/homework/divisibility/lessons/Finding-Greatest-Common-Divisor-of-integer-numbers.lesson>Finding Greatest Common Divisor of integer numbers</A> - <A HREF=http://www.algebra.com/algebra/homework/divisibility/lessons/Relativity-prime-numbers-help-to-solve-the-problem.lesson>Relatively prime numbers help to solve the problem</A> - <A HREF=https://www.algebra.com/algebra/homework/divisibility/lessons/Quadratic-polynomial-with-odd-integer-coefs-can-not-have-a-rational-root.lesson>Quadratic polynomial with odd integer coefficients can not have a rational root</A> - <A HREF=https://www.algebra.com/algebra/homework/divisibility/Proving-the-equation-has-no-integer-solutions.lesson>Proving an equation has no integer solutions</A> - <A HREF=https://www.algebra.com/algebra/homework/divisibility/lessons/Composite-number-of-the-form-%284n%2B3%29-must--have-a-prime-divisor-of-the-form-%284n%2B3%29.lesson>Composite number of the form (4n+3) must have a prime divisor of the form (4n+3)</A> - <A HREF=https://www.algebra.com/algebra/homework/divisibility/lessons/Problems-on-divisors-of-a-given-number.lesson>Problems on divisors of a given number</A> - <A HREF=https://www.algebra.com/algebra/homework/divisibility/lessons/How-many-three-digit-numbers-are-multiples-of-both-5-and-7.lesson>How many three-digit numbers are multiples of both 5 and 7?</A> - <A HREF=https://www.algebra.com/algebra/homework/divisibility/lessons/How-many-3-digit-numbers-are-not-dvsbl-by-2-not-dvsbl-by-3-notdvsbl-by-either-2-or-3.lesson>How many 3-digit numbers are not divisible by 2; not divisible by 3; not divisible by either 2 or 3</A> - <A HREF=https://www.algebra.com/algebra/homework/divisibility/lessons/How-many-integer-numbers-in-the-range-1-300-are-divisible-by.lesson>How many integer numbers in the range 1-300 are divisible by at least one of the integers 4, 6 and 15 ?</A> - <A HREF=https://www.algebra.com/algebra/homework/divisibility/lessons/Find-the-remainder-of-division.lesson>Find the remainder of division </A> - <A HREF=https://www.algebra.com/algebra/homework/divisibility/lessons/Why-3%5En%2B7%5En-2-is-divisible-by-8-for-all-positive-integer-n.lesson>Why 3^n + 7^n - 2 is divisible by 8 for all positive integer n ?</A> - <A HREF=https://www.algebra.com/algebra/homework/divisibility/What-is--the-last-digit-of-the-number-a%5En.lesson>What is the last digit of the number a^n ?</A> - <A HREF=https://www.algebra.com/algebra/homework/divisibility/lessons/Find-the-last-three-digits-of-these-numbers.lesson>Find the last three digits of these numbers</A> - <A HREF=https://www.algebra.com/algebra/homework/divisibility/lessons/Find-the-two-last-digits-of-the-number-3%5E123%2B7%5E123%2B9%5E123.lesson>Find the last two digits of the number 3^123 + 7^123 + 9^123</A> - <A HREF=https://www.algebra.com/algebra/homework/divisibility/Find-the-last-two-digits-of-%281%21-%2B-2%21-%2B-3%21-%2B-%2B-2024%21%29%5E2024.lesson>Find the last two digits of (1! + 2! + 3! + ... + 2024!)^2024</A> - <A HREF=https://www.algebra.com/algebra/homework/divisibility/lessons/Find-n-th-term-of-a-sequence.lesson>Find n-th term of a sequence</A> - <A HREF=https://www.algebra.com/algebra/homework/divisibility/lessons/Solving-one-Diofantine-equation.lesson>Solving Diophantine equations</A> - <A HREF=https://www.algebra.com/algebra/homework/divisibility/lessons/Determine-the-number-of-integer-solutions-to-equation-n%5E2%2B18n%2B3=m%5E2.lesson>How many integers of the form n^2 + 18n + 13 are perfect squares</A> - <A HREF=https://www.algebra.com/algebra/homework/divisibility/lessons/Miscellaneous-problems-on-divisibility-numbers.lesson>Miscellaneous problems on divisibility numbers</A> - <A HREF=https://www.algebra.com/algebra/homework/divisibility/lessons/Find-the-sum-of-digits-of-integer-numbers.lesson>Find the sum of digits of integer numbers</A> - <A HREF=https://www.algebra.com/algebra/homework/divisibility/lessons/Two-digit-numbers-with-digit-9.lesson>Two-digit numbers with digit "9"</A> - <A HREF=https://www.algebra.com/algebra/homework/divisibility/lessons/Find-a-triangle-with-integer-side-lengths-and-integer-area.lesson>Find a triangle with integer side lengths and integer area</A> - <A HREF=https://www.algebra.com/algebra/homework/divisibility/lessons/Math-circle-level-problem-on-the-hundred-handed-monster-Briareus.lesson>Math circle level problem on the hundred-handed monster Briareus</A> - <A HREF=https://www.algebra.com/algebra/homework/divisibility/lessons/Math-Citcle-level-problem-on-lockers-and-divisors.lesson>Math Circle level problem on lockers and divisors of integer numbers</A> - <A HREF=https://www.algebra.com/algebra/homework/divisibility/lessons/Nice-entertaiment-problem-What-numbers-John-is-thinking-about.lesson>Nice entertainment problems related to divisibility property</A> - <A HREF=https://www.algebra.com/algebra/homework/divisibility/lessons/Solving-problems-on-modular-arithmetic.lesson>Solving problems on modular arithmetic</A> - <A HREF=https://www.algebra.com/algebra/homework/divisibility/lessons/Using-the-little-Fermat-theorem-to-solve-a-problem-on-modular-arithmetic.lesson#google_vignette>Using the little Fermat's theorem to solve a problem on modular arithmetic</A> - <A HREF=https://www.algebra.com/algebra/homework/divisibility/lessons/OVERVIEW-of-miscellanious-solved-problems-on-divisibility-of-numbers.lesson>OVERVIEW of miscellaneous solved problems on divisibility of integer numbers</A>
