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Question Number 147585 by mathdanisur last updated on 22/Jul/21

	Answered by Rasheed.Sindhi last updated on 22/Jul/21
	$z=w+f : whole number & fraction<1 w+f−2021f=2021 w=2021f−f+2021 w=2020f+2021 Let f=(p/q) ; p & q coprime p<q w=2020((p/q))+2021 z=w+f=2020((p/q))+2021+(p/q) z=2021((p/q))+2021 z=2021(((p+q)/q)) q ∣ 2020 q=2⇒p=1⇒f=(1/2)⇒w=2020((1/2))+2021 w=3031⇒z=3031(1/2) q=4⇒p=1,3⇒f=(1/4),(3/4)⇒w=2020((1/4))+2021 w=2526⇒z=2526(1/4) w=2020((3/4))+2021=3536⇒z=3536(3/4) q=5⇒p=1,2,3,4⇒f=(1/5),(2/5),(3/5),(4/5) w=2020((1/5))+2021=2425 z=2425(1/5) z=2021(((p+q)/q)) Where q ∣ 2020 ∧ (p,q)=1 ∧ p<q$
	$z = w + f : w h o l e n u m b e r & f r a c t i o n < 1$ $w + f - 2021 f = 2021$ $w = 2021 f - f + 2021$ $w = 2020 f + 2021$ $L e t f = \frac{p}{q}; p & q c o p r i m e p < q$ $w = 2020 (\frac{p}{q}) + 2021$ $z = w + f = 2020 (\frac{p}{q}) + 2021 + \frac{p}{q}$ $z = 2021 (\frac{p}{q}) + 2021$ $z = 2021 (\frac{p + q}{q})$ $q ∣ 2020$ $q = 2 \Rightarrow p = 1 \Rightarrow f = \frac{1}{2} \Rightarrow w = 2020 (\frac{1}{2}) + 2021$ $w = 3031 \Rightarrow z = 3031 \frac{1}{2}$ $q = 4 \Rightarrow p = 1, 3 \Rightarrow f = \frac{1}{4}, \frac{3}{4} \Rightarrow w = 2020 (\frac{1}{4}) + 2021$ $w = 2526 \Rightarrow z = 2526 \frac{1}{4}$ $w = 2020 (\frac{3}{4}) + 2021 = 3536 \Rightarrow z = 3536 \frac{3}{4}$ $q = 5 \Rightarrow p = 1, 2, 3, 4 \Rightarrow f = \frac{1}{5}, \frac{2}{5}, \frac{3}{5}, \frac{4}{5}$ $w = 2020 (\frac{1}{5}) + 2021 = 2425$ $z = 2425 \frac{1}{5}$ $z = 2021 (\frac{p + q}{q})$ $W h e r e q ∣ 2020 \land (p, q) = 1 \land p < q$
	Commented by mathdanisur last updated on 22/Jul/21

	$t h a n k y o u S e r$ $a n s w e r z = 2021 + n + n / 2020$ $n = 0; 1; 2; . . . 2020$
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