Let-X-is-natural-number-Multification-of-its-digits-is-X-2-97X-2020-Find-the-value-of-X-maximum-X-minimum-

Question Number 25230 by naka3546 last updated on 06/Dec/17

L e t X i s n a t u r a l n u m b e r .

M u l t i f i c a t i o n o f i t s d i g i t s i s X^{2} - 97 X + 2020 .

F i n d t h e v a l u e o f X_{m a x i m u m} + X_{m i n i m u m} .

Answered by Rasheed.Sindhi last updated on 07/Dec/17

Let X = {(x_{n - 1} x_{n - 2} \dots x_{1} x_{0})}_{10}, where x_{n - 1},

x_{n - 2}, \dots, x_{2}, x_{1} are decimal digits .

X = x_{n - 1} {(10)}^{n - 1} + x_{n - 2} {(10)}^{n - 2} + \dots x_{1} {(10)}^{1} + x_{0}

x_{n - 1} . x_{n - 2} \dots x_{1} . x_{0} = X^{2} - 97 X + 2020

An n - digit {number}_{\max} = \overset{n nines}{99 \dots 9}

So \overset{multiplication of digis}{X^{2} - 97 X + 2020} ⩽ 9^{n}

For 1 - digit X^{2} - 97 X + 2020 ⩽ 9

X ⩽ 66, X ⩽ 30 \Rightarrow X ⩽ 66

The number is upto 2 digit

For 2 - digit X^{2} - 97 X + 2020 ⩽ 81

X ⩽ 28, X ⩽ 68 \Rightarrow X ⩽ 68 \dots (i)

The number is upto 2 digit .

For 3 - digit X^{2} - 97 X + 2020 ⩽ 729

X ⩽ 15, X ⩽ 81 \Rightarrow X ⩽ 81

The number cannot be 3 digit .

For 4 - digit X^{2} - 97 X + 2020 ⩽ 6561

X ⩽ - 34, X ⩽ 131 \Rightarrow X ⩽ 131

The number cannot be 4 digit .

\dots \dots

\dots .

. .

It seems that the number cannot

consist of more than 2 - digit .

If X is two digit number .

Let the number is 10 a + b

ab = {(10 a + b)}^{2} - 97 (10 a + b) + 2020

\Rightarrow {100 a}^{2} + 19 ab + b^{2} - 970 a - 97 b + 2020 = 0 \dots (ii)

Also from (i)

10 a + b ⩽ 68 \Rightarrow 1 ⩽ a ⩽ 6 \land 0 ⩽ b ⩽ 8

a = 1 \Rightarrow

(ii) \Rightarrow 100 + 19 b + b^{2} - 970 - 97 b + 2020 = 0

b^{2} - 78 b + 1150 = 0 [No integer b]

a = 1, 2, 3, 4, 5 \Rightarrow value of b is inadmissble

But for a = 6 \Rightarrow

(ii) \Rightarrow 3600 + 114 b + b^{2} - 5820 - 97 b + 2020 = 0

b^{2} + 17 b - 200 = 0 \Rightarrow b = 8

Hence the single 2 - digit number is 68

Verify that

6 \times 8 = 68^{2} - 97 (68) + 2020

X_{\max} + X_{\min} = 68 + 68 = 136

Commented by Rasheed.Sindhi last updated on 07/Dec/17

^{∙} Sir mrW 1

Thank S s s s for correction .

^{∙} Mr . naka,

I think {there}^{'} s only one such number

X = 68

6 \times 8 = 68^{2} - 97 (68) + 2020

? please confirm this .

I will try to complete my answer

after your confirmation .

Commented by mrW1 last updated on 07/Dec/17

s i r, p l e a s e c h e c k :

f o r n - d i g i t n u m b e r s

So \overset{multiplication of digis}{X^{2} - 97 X + 2020} ⩽ 9^{n}

Commented by mrW1 last updated on 07/Dec/17

g r e a t w o r k!

Commented by Rasheed.Sindhi last updated on 07/Dec/17

THANKS a LOT S i r!