Question

a*b=a+b+2020 aflati a si b
$a \times b = a + b + 2020$
​

Answer 1

ab=a+b+2020

ab-a=b+2020

a(b-1)=b+2020

a=(b+2020)/(b-1)=(b-1)/(b-1)+2021/(b-1)=1+2021/(b-1)

=>b-1|2021

D2021={-2021,-43,-47,-1,1,43,47,2021}

b-1=-2021=>b=-2020; b-1=-47 => b=-46

b=-42; b=0; b=2; b=44; b=48; b=2022

Pentru: b=-2020=>a=0

b=-46=>a=-42

b=-42=>a=-46

b=0=>a=-2020

b=2=>a=2022

b=44=>a=48

b=48=>a=44

b=2022=>a=2

( am inlocuit in a=(b+2020)/(b-1) )

Answer 2

a×b = a+b + 2020

⇔ a×b - a - b = 2020

⇔ b(a-1) - a = 2020

⇔ b(a-1) - a+1 = 2021

⇔ b(a-1) - (a-1) = 2021

⇔ (a-1)(b-1) = 2021

2021 = 1×43×47

⇒ 43×47 = 2021 ⇒ {a = 44,  b = 48}

⇔ (-43)×(-47) = 2021 ⇒ {a = -42,  b = -46}

⇔ 47×43 = 2021 ⇒ {a = 48,  b = 44}

⇔ (-47)×(-43) = 2021 ⇒ {a = -46,  b = -42}

⇔ 2021×1 = 2021 ⇒ {a = 2022,  b = 2}

⇔ (-2021)×(-1) = 2021 ⇒ {a = -2020,  b = 0}

⇔ 1×2021 = 2021 ⇒ {a = 2,  b = 2022}

⇔ (-1)×(-2021) = 2021 ⇒ {a = 0,  b = -2020}