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For the given pair (x,y) of positive integers, such that 4x−17y=1 and x<1000 how many integer values of y satisfy the given conditions?
Correct answer is '59'. Can you explain this answer?
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For the given pair (x,y) of positive integers, such that 4x−17y=...
We first need to find out a solution for x and y. Once we get a solution, values of x would be in an AP with a common difference of 17 whereas values of y would be in an AP with a common difference of 4
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For the given pair (x,y) of positive integers, such that 4x−17y=...
Since $4x
Now, we want to find the number of values of $y$ such that there exist positive integers $x$ satisfying the given conditions. We can rephrase this as finding the number of integers $y$ such that $\left\lfloor\frac{y}{4}\right\rfloor\geq 1$.
Let $4k\leq y<4k+4$ for="" some="" non-negative="" integer="" $k$.="" then="" $\left\lfloor\frac{y}{4}\right\rfloor="k$." thus,="" the="" number="" of="" integers="" $y$="" such="" that="" $\left\lfloor\frac{y}{4}\right\rfloor\geq="" 1$="" is="" equal="" to="" the="" number="" of="" non-negative="" integers="" $k$="" such="" that="" $4k+4\leq="" 2016$.="" solving="" the="" inequality,="" we="" get="" $k\leq="" 503$.="" thus,="" there="" are="" $504$="" possible="" values="" of="" $k$,="" and="" therefore="" $\boxed{504}$="" possible="" values="" of="" $y$.="" for="" some="" non-negative="" integer="" $k$.="" then="" $\left\lfloor\frac{y}{4}\right\rfloor="k$." thus,="" the="" number="" of="" integers="" $y$="" such="" that="" $\left\lfloor\frac{y}{4}\right\rfloor\geq="" 1$="" is="" equal="" to="" the="" number="" of="" non-negative="" integers="" $k$="" such="" that="" $4k+4\leq="" 2016$.="" solving="" the="" inequality,="" we="" get="" $k\leq="" 503$.="" thus,="" there="" are="" $504$="" possible="" values="" of="" $k$,="" and="" therefore="" $\boxed{504}$="" possible="" values="" of="">
Now, we want to find the number of values of $y$ such that there exist positive integers $x$ satisfying the given conditions. We can rephrase this as finding the number of integers $y$ such that $\left\lfloor\frac{y}{4}\right\rfloor\geq 1$.
Let $4k\leq y<4k+4$ for="" some="" non-negative="" integer="" $k$.="" then="" $\left\lfloor\frac{y}{4}\right\rfloor="k$." thus,="" the="" number="" of="" integers="" $y$="" such="" that="" $\left\lfloor\frac{y}{4}\right\rfloor\geq="" 1$="" is="" equal="" to="" the="" number="" of="" non-negative="" integers="" $k$="" such="" that="" $4k+4\leq="" 2016$.="" solving="" the="" inequality,="" we="" get="" $k\leq="" 503$.="" thus,="" there="" are="" $504$="" possible="" values="" of="" $k$,="" and="" therefore="" $\boxed{504}$="" possible="" values="" of="" $y$.="" for="" some="" non-negative="" integer="" $k$.="" then="" $\left\lfloor\frac{y}{4}\right\rfloor="k$." thus,="" the="" number="" of="" integers="" $y$="" such="" that="" $\left\lfloor\frac{y}{4}\right\rfloor\geq="" 1$="" is="" equal="" to="" the="" number="" of="" non-negative="" integers="" $k$="" such="" that="" $4k+4\leq="" 2016$.="" solving="" the="" inequality,="" we="" get="" $k\leq="" 503$.="" thus,="" there="" are="" $504$="" possible="" values="" of="" $k$,="" and="" therefore="" $\boxed{504}$="" possible="" values="" of="">
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For the given pair (x,y) of positive integers, such that 4x−17y=1 and x<1000 how many integer values of y satisfy the given conditions?Correct answer is '59'. Can you explain this answer?
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For the given pair (x,y) of positive integers, such that 4x−17y=1 and x<1000 how many integer values of y satisfy the given conditions?Correct answer is '59'. Can you explain this answer? for CAT 2025 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about For the given pair (x,y) of positive integers, such that 4x−17y=1 and x<1000 how many integer values of y satisfy the given conditions?Correct answer is '59'. Can you explain this answer? covers all topics & solutions for CAT 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for For the given pair (x,y) of positive integers, such that 4x−17y=1 and x<1000 how many integer values of y satisfy the given conditions?Correct answer is '59'. Can you explain this answer?.
For the given pair (x,y) of positive integers, such that 4x−17y=1 and x<1000 how many integer values of y satisfy the given conditions?Correct answer is '59'. Can you explain this answer? for CAT 2025 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about For the given pair (x,y) of positive integers, such that 4x−17y=1 and x<1000 how many integer values of y satisfy the given conditions?Correct answer is '59'. Can you explain this answer? covers all topics & solutions for CAT 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for For the given pair (x,y) of positive integers, such that 4x−17y=1 and x<1000 how many integer values of y satisfy the given conditions?Correct answer is '59'. Can you explain this answer?.
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