Determine-m-amp-n-such-that-digit-sum-m-2-n-amp-digit-sum-n-2-m-digit-sum-abc-a-b-c-

Question Number 168937 by Rasheed.Sindhi last updated on 21/Apr/22

\underset{―}{D e t e r m i n e m & n s u c h t h a t :}

\underset{\underset{d i g i t - s u m (n^{2}) = m}{&}}{d i g i t - s u m (m^{2}) = n}

^{} d i g i t - s u m (\overset{―}{a b c . .} .) = a + b + c + \dots

Answered by mindispower last updated on 22/Apr/22

s u p p o s e m h a s e a d i g i t s, m ⩾ n

a n d n h a s e b d i g i t s n

a ⩾ b

\Rightarrow m^{2} ⩽ 10^{2 a} \Rightarrow s u m - d i g i t m^{2} ⩽ 9.2 a = 18 a

n^{2} ⩽ 10^{2 b} \Rightarrow S u m - d i g i t s n^{2} ⩽ 18 b

n ⩽ m ⩽ 18 b ⩽ 18 a

m ⩾ 10^{a - 1} \Rightarrow 10^{a - 1} ⩽ 18 a

a ⩽ 2 \Rightarrow m ⩽ 36

a = 1 \Rightarrow d i g i t s u m m^{2} ⩽ 13

d i g i t - s u m n^{2} ⩽ 9

b ⩽ a \Rightarrow b = 1,

n \in {0, 1, 2, 3, 4, 5, 6, 8, 9}

n = 1 \Rightarrow m = 1, n = m = 0

a = 2 t o o b e e c o n t i n u e d