Lesson Nice entertainment problems related to divisibility property
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<H2>Nice entertainment problems related to divisibility property</H2> <H3>Problem 1</H3>John thinks of two positive integers. He multiplies them together and then subtracts each of the integers from the product, with a result of 35. Find all possible pairs of numbers he could have chosen. <B>Solution</B> <pre> Let x and y be two positive integers John thinks about. Then from the condition, you have this equation xy - x - y = 35. Add 1 (one) to both side. You will get xy - x - y + 1 = 36. It is equivalent to (x-1)*(y-1) = 36 Thus the integer number 36 is presented as the product of integer positive factors (x-1) and (y-1). The possible factors of the number 36 are 1, 2, 3, 4, 6, 9, 12, 18, 36. So, (x-1) may have these values 1, 2, 3, 4, 6, 9, 12, 18, 36 (and, correspondingly, (y-1) has the values of the same list in reversed order). So, all possible values for numbers x and y are these pairs (x,y) = (2,37), (3,19), (4,13), (7,7), (10,5), (13,4), (19,3) and (37,2). <U>ANSWER</U> </pre> <H3>Problem 2</H3>Three girls, Ann, Betty and Cynthia, each have a younger brother, Dylan, Ernie and Frank, respectively. All six children do some fruit picking for the local farmer. The farmer agrees to pay each child as many dollars per basket as the number of baskets of fruit collected by that child. Each of the girls earned $45 more than her brother, and all six children collected a different number of baskets. How much did the farmer pay them all in total ? <B>Solution</B> <pre> Let x be the number of baskets collected by some girl, and y be the number of basket collected by her brother. Then for each such a pair, we have this equation x^2 - y^2 = 45. Factor left side to get (x-y)*(x+y) = 45. 45 has these decompositions in the product of factors 1*45, 3*15 and 9*5. These decompositions produce these systems of equations a) x - y = 1 x + y = 45 with the solution x = 23, y = 22; b) x - y = 3 x + y = 15 with the solution x = 9, y = 6; c) x - y = 5 x + y = 9 with the solution x = 7, y = 2. These are all possible solutions to the numbers of baskets collected by the children. There is NO other solutions. Hence, the total pay was {{{23^2 + 22^2 + 9^2 + 6^2 + 7^2 + 2^2}}} = 1183 dollars. <U>ANSWER</U> </pre> My other additional lessons on Miscellaneous word problems in this site are - <A HREF=https://www.algebra.com/algebra/homework/word/misc/I-do-not-have-enough-savings-now.lesson>I do not have enough savings now</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/In-a-jar-all-but-6-are-red-marbles.lesson>In a jar, all but 6 are red marbles</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/How-many-boys-and-how-many-girls-are-there-in-a-family.lesson>How many boys and how many girls are there in a family ?</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/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/word/misc/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/word/misc/What-are-the-two-last-digits-of-the-number-3%5E123%2B7%5E123%2B9%5E123.lesson>What are the last two digits of the number 3^123 + 7^123 + 9^123 ?</A> - 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