Lesson Relatively prime numbers help to solve problems
Algebra
->
Divisibility and Prime Numbers
->
Lessons
-> Lesson Relatively prime numbers help to solve problems
Log On
Algebra: Divisibility and Prime Numbers
Section
Solvers
Solvers
Lessons
Lessons
Answers archive
Answers
Source code of 'Relatively prime numbers help to solve problems'
This Lesson (Relatively prime numbers help to solve problems)
was created by by
ikleyn(52908)
:
View Source
,
Show
About ikleyn
:
<H2>Relatively prime numbers help to solve problems</H2> <H3>Problem 1</H3>A group of boys and girls sit a test. Exactly 2/3 of the boys and 3/4 of the girls pass the test. If an equal number of boys and girls passed the test, what fraction of the entire group passed the test? <B>Solution</B> <pre> Let b = the number of boys and g = the number of girls in the class. Then we have {{{(2/3)*b}}} = {{{(3/4)*g}}}, according to the condition. Let us write it using the common denominator: {{{(8/12)*b}}} = {{{(9/12)*g}}}, or, which is the same, {{{(8b)/12}}} = {{{(9g)/12}}}. It implies that 8b = 9g. (1) Next, since the numbers 8 and 9 are <U>relatively primes</U>, (1) implies that b is multiple of 9 and g is multiple of 8: b = 9*n, g = 8*m (2) with integer n and m. Then you can re-write (1) in the form 8*9n = 9*8m, or 72n = 72m. It implies that n = m and, hence, b = 9n, g = 8n (3) with some integer n. Now, the number of those students who passed the test is {{{(2/3)*b + (3/4)*g}}} = {{{(8b)/12 + (9g)/12}}} = {{{(8*9n)/12 + (9*8n)/12}}} = {{{(72n)/12 + (72*n)/12}}} = {{{(144/12)*n}}} = 12n. (5) The total number of students in the class is b + g = 9n + 8n = 17n. (6) Now it is easy to calculate the ratio of those who passed the test to the total number of students in the class. It is (5) divided by (6): {{{(12n)/(17n)}}} = {{{12/17}}}. <U>Answer</U>. The ratio of those who passed the test to the total number of students in the class is {{{12/17}}}. </pre> <H3>Problem 2</H3>The numerator of a fraction is increased by 8 and the denominator is decreased by 1, the resulting fraction is the reciprocal of the original fraction. What is the original fraction? <B>Solution</B> <pre> The basic equation is {{{(n+8)/(d-1)}}} = {{{d/n}}}. It implies after cross multiplying n^2 + 8n = d^2 -d ====> n^2 - d^2 = -8n - d ====> (n+d)*(n-d) = -8n - d ====> (n+d)*(d-n) = 8n + d ====> (n+d)*(d-n) = (8n - 8d) + (8d + d) ====> (n+d)*(d-n) = -8*(d-n) + 9d ====> divide both sides by (d-n) ====> n + d = -8 + {{{(9d)/(d-n)}}}. (1) By the way (as an aside notice) it follows from the last formula that d > n. Now, from the very beginning we can assume that our ratio/fraction is just REDUCED, so n and d have no common factors and are relatively prime numbers. Then "d" and "d-n" in (1) are relatively prime TOO. Then, since {{{(9d)/(d-n)}}} in (1) is an integer number, it implies that (d-n) divides 9. So, the only possible cases are a) d-n = 1; b) d-n = 3; c) d-n = 9. Case a) d-n = 1 implies (through (1)) that n+d = -8 + 9d. Then you have this system of two eqns d-n = 1 and n+d = -8+9d. It has the only solution n=0, d=1, but this solution DOESN't work for global problem. case b) d-n = 3 implies (through (1) ) n+d = -8 + {{{(9*d)/3}}} = -8+3d. Then you have this system of two eqns d-n = 3 and n+d = -8+3d. It has the only solution n=2, d=5. Then the fraction is {{{n/d}}} = {{{2/5}}}, and you can easily check that it works for the global problem. Case c) d-n = 9 implies (through (1) ) n+d = -8 + {{{(9*d)/9}}} = -8+d. Then you have this system of two eqns d-n = 9 and n+d = -8+d. but this solution DOESN't work for global problem. </pre> Thus the only solution for global problem is n = 2, d = 5 with the fraction 2/5 = {{{2/5}}}. 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=https://www.algebra.com/algebra/homework/divisibility/lessons/Solving-equations-in-integer-numbers.lesson>Solving equations in integer numbers</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</U>