Find-distinct-natural-numbers-from-1-to-9-such-that-these-six-equations-are-satisfied-simultaneously-1-a-bc-20-2-d-e-f-20-3-g-hi-20-4-adg-20-5-b-eh-20-6-c-f-i-1

Question Number 16701 by Tinkutara last updated on 25/Jun/17

Find distinct natural numbers from 1

to 9 such that these six equations are

satisfied simultaneously :

(1) a + b c = 20

(2) d + e + f = 20

(3) g - h i = - 20

(4) a d g = 20

(5) b + e h = 20

(6) c + f - i = 10

Answered by RasheedSoomro last updated on 26/Jun/17

{a, b, c, d, e, f, g, h, i} = {1, 2, 3, 4, 5, 6, 7, 8, 9}

adg = 20 = 1 \times 4 \times 5 [Only one way disregard to order]

∴ {a, d, g} = {1, 4, 5} \dots \dots \dots \dots \dots \dots \dots \dots (i)

∴ {b, c, e, f, h, i} = {2, 3, 6, 7, 8, 9}

{\begin{cases} a = 1 \Rightarrow a + bc = 20 \Rightarrow bc = 19 \neq (1 ⩽ x ⩽ 9) \times (1 ⩽ y ⩽ 9), ∴ a \neq 1 \\ a = 4 \Rightarrow bc = 16 = 2 \times 8 \Rightarrow {b, c} = {2, 8} \\ a = 5 \Rightarrow bc = 15 = 3 \times 5 but b ∣ c \neq 5 because a = 5 \end{cases}

So, surely a = 4 \land {b, c} = {2, 8}

∴ {e, f, h, i} = {3, 6, 7, 9}

{\begin{cases} g \neq 4 because a = 4 \\ g = 1 \Rightarrow hi - g = 20 \Rightarrow hi = 21 = 3 \times 7 \Rightarrow {h, i} = {3, 7} \\ g = 5 \Rightarrow hi = 25 = 5 \times 5 but {g, h, i} \neq {5, 5, 5} \end{cases}

So, surely g = 1 \land {h, i} = {3, 7}

∴ {e, f} = {6, 9}

{\begin{cases} b = 2 \Rightarrow b + eh = 20 \Rightarrow eh = 18 = 3 \times 6 \Rightarrow {e, h} = {3, 6} \\ b = 8 \Rightarrow eh = 12^{∙} \end{cases}

^{∙} 12 can be 2 \times 6 or 3 \times 4 but these values are assigned to others

{a, d, g} = {1, 4, 5}, a = 4, g = 1 \Rightarrow d = 5

\begin{matrix} {h, i} = {3, 7} \\ {e, h} = {3, 6} \\ {e, f} = {6, 9} \end{matrix}} \Rightarrow h = 3, i = 7, e = 6, f = 9

{b, c} = {2, 8}

{\begin{cases} c + f - i = 10 \Rightarrow c + 9 - 7 = 10 \Rightarrow c = 8 \\ {b, c} = {2, 8} \land c = 8 \Rightarrow b = 2 \end{cases}

a = 4, b = 2, c = 8, d = 5, e = 6, f = 9, g = 1, h = 3, i = 7

Commented by Tinkutara last updated on 26/Jun/17

Thanks Sir!