N=x32y in base 5. determinate x and y such that N is divisible by 3 and 4.


Question and Answers Forum

All Questions Topic List
Algebra Questions
Previous in All Question Next in All Question
Previous in Algebra Next in Algebra
Question Number 124109 by mathocean1 last updated on 30/Nov/20

$N = x 32 y i n b a s e 5 .$ $d e t e r m i n a t e x a n d y s u c h t h a t N i s$ $d i v i s i b l e b y 3 a n d 4 .$
	Answered by floor(10²Eta[1]) last updated on 03/Dec/20

	$1 ⩽ x ⩽ 4 and 0 ⩽ y ⩽ 4$ $N = 5^{3} x + 3 . 5^{2} + 2.5 + y$ $N \equiv 0 (\mod 3) and N \equiv 0 (\mod 4)$ $\Rightarrow N \equiv - x + 3 - 2 + y = - x + y + 1 \equiv 0 (\mod 3)$ $\Rightarrow N \equiv x + 3 + 2 + y \equiv x + y + 1 (\mod 4)$ $3 a = - x + y + 1$ $- 3 ⩽ - x + 1 ⩽ 3 a = - x + y + 1 ⩽ - x + 5 ⩽ 4$ $- 3 ⩽ 3 a ⩽ 4 \Rightarrow - 1 ⩽ a ⩽ 1 \Rightarrow a = 1, a = 0, a = - 1$ $4 b = x + y + 1$ $2 ⩽ y + 2 ⩽ 4 b = x + y + 1 ⩽ y + 5 ⩽ 9$ $2 ⩽ 4 b < 8 \Rightarrow 0.5 ⩽ b ⩽ 1 \Rightarrow b = 1$ $x + y + 1 = 4 \Rightarrow x + y = 3$ $- x + y + 1 = 3 or - x + y + 1 = 0 or - x + y + 1 = - 3$ $- x + y = 2 or - x + y = - 1 or - x + y = - 4$ $y = \frac{5}{2} \notin N or y = 1 \Rightarrow x = 2 or y = - \frac{1}{2} \notin N$ $N = 2321_{5}$
	Commented by floor(10²Eta[1]) last updated on 03/Dec/20

	$3 a = - x + y + 1$ $0 ⩽ y ⩽ 4$ $\Rightarrow - x + y + 1 ⩽ - x + 4 + 1 = - x + 5$ $\Rightarrow - x + y + 1 ⩾ - x + 1$ $\Rightarrow - x + 1 ⩽ 3 a ⩽ - x + 5$ $but 1 ⩽ x ⩽ 4 \Rightarrow - 1 ⩾ - x ⩾ - 4$ $\Rightarrow - x + 1 ⩾ - 4 + 1 = - 3$ $\Rightarrow - x + 5 ⩽ - 1 + 5 = 4$ $\Rightarrow - 3 ⩽ - x + 1 ⩽ 3 a ⩽ - x + 5 ⩽ 4$ $\Rightarrow - 3 ⩽ 3 a ⩽ 4$ $- 1 ⩽ a ⩽ 1$ $a = - 1, a = 0, a = 1$
	Commented by mathocean1 last updated on 03/Dec/20

	$P l e a s e s i r c a n y o u e x p l a i n t h e 7^{t h} l i n e$
	Commented by mathocean1 last updated on 03/Dec/20

	$t h a n k v e r y m u c h s i r$
Terms of Service
Privacy Policy
Contact: info@tinkutara.com

Question and Answers Forum