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Omni-MATH / For positive reals p and q, define the remainder when p is divided by q as the smallest nonnegative real r such that (p−r)/(q) is an integer. For an ordered pair (a,b) of positive …

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problem

For positive reals pp and qq, define the remainder when pp is divided by qq as the smallest nonnegative real rr such that p−rq\frac{p-r}{q} is an integer. For an ordered pair (a,b)(a, b) of positive integers, let r1r_{1} and r2r_{2} be the remainder when a2+b3a \sqrt{2}+b \sqrt{3} is divided by 2\sqrt{2} and 3\sqrt{3} respectively. Find the number of pairs (a,b)(a, b) such that a,b≤20a, b \leq 20 and r1+r2=2r_{1}+r_{2}=\sqrt{2}.
Plain-text mathematical notation (without MathML)
For positive reals p and q, define the remainder when p is divided by q as the smallest nonnegative real r such that (p−r)/(q) is an integer. For an ordered pair (a,b) of positive integers, let r₁ and r₂ be the remainder when a√(2)+b√(3) is divided by √(2) and √(3) respectively. Find the number of pairs (a,b) such that a,b≤20 and r₁+r₂=√(2).
Original LaTeX notation
For positive reals $p$ and $q$, define the remainder when $p$ is divided by $q$ as the smallest nonnegative real $r$ such that $\frac{p-r}{q}$ is an integer. For an ordered pair $(a, b)$ of positive integers, let $r_{1}$ and $r_{2}$ be the remainder when $a \sqrt{2}+b \sqrt{3}$ is divided by $\sqrt{2}$ and $\sqrt{3}$ respectively. Find the number of pairs $(a, b)$ such that $a, b \leq 20$ and $r_{1}+r_{2}=\sqrt{2}$.

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Source and history

task_id
3ba2707f-8bae-5654-8c80-c3767cadbc29
task_key
test--3ba2707f-8bae-5654-8c80-c3767cadbc29
task_revision_id
1
config / split
/ test

Official source

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