Solve-in-N-6-x-6-y-222-

Question Number 172210 by Rasheed.Sindhi last updated on 24/Jun/22

Solve in N :

6^{x} + 6^{y} = 222

Commented by infinityaction last updated on 24/Jun/22

(1, 3) a n d (3, 1)

Commented by Rasheed.Sindhi last updated on 24/Jun/22

Correct sir! Process please!

Commented by infinityaction last updated on 22/Jul/22

x, y \in N

6^{x} + 6^{y} = 222

6^{4} = 1296 ⪈ 222

s o x a n d y ⪇ 4

a n d x a n d y ⪈ 0

0 ⪇ x, y ⪇ 4

s o x a n d y s h o u l d b e 1, 2, 3

s o c o m b i t i o n s o f n u m b e r s

(x, y) = {(1, 3) (3, 1)}

Commented by Rasheed.Sindhi last updated on 24/Jun/22

T h a n X s i r!

Answered by Rasheed.Sindhi last updated on 24/Jun/22

A n O t h e r W a y

6^{x} + 6^{y} = 222

\Rightarrow 6^{x - 1} + 6^{y - 1} = 37

37 i s s u m o f t w o p o w e r s o f 6 a n d

o n e i s c e r t a i n l y o d d (6^{0} i s o n l y o d d)

∴ 7 = 36 + 1 ∣ 7 = 1 + 36

▸ 6^{x - 1} + 6^{y - 1} = 36 + 1 = 6^{2} + 6^{0}

x - 1 = 2 \land y - 1 = 0

x = 3 \land y = 1

▸ 6^{x - 1} + 6^{y - 1} = 1 + 36 = 6^{0} + 6^{2}

x - 1 = 0 \land y - 1 = 2

x = 1 \land y = 3

(x, y) = (1, 3), (3, 1)

Commented by infinityaction last updated on 24/Jun/22

n i c e s i r

Commented by Rasheed.Sindhi last updated on 24/Jun/22

Answered by kapoorshah last updated on 24/Jun/22

6^{x - y} . 6^{y} + 6^{y} = 222

6^{y} (6^{x - y} + 1) = 222

6^{y} (6^{x - y} + 1) = 6 . 37

6^{y} = 6^{1} \Rightarrow y = 1

6^{x - 1} + 1 = 37

6^{x - 1} = 6^{2} \Rightarrow x = 3

s o t h e s o l u t i o n s : x = 3, y = 1 a n d

x = 1, y = 3

Commented by Rasheed.Sindhi last updated on 24/Jun/22

T han k s sir!