Question

If abc + abc + abc = ccc
Given :
- that a,b and c are single digit natural numbers.
- abc and ccc represents a 3 digit number

Find the value of (a*b*c)- (a*b) – (a*c) - a - b - c ?

Answer 1

Given,

a, b, c are singular digits

abc is a 3 digit number.

ccc is a 3 digit number.

abc + abc + abc = ccc.... (1)

We have,

From (1),

3*abc = ccc

We know,

abc and ccc are 3 digit numbers.

A 3 digit number is written like = 100a + 10b + c

So again,

From (1),

3{100a + 10b + c} = 100c + 10c + c

300a + 30b + 3c = 100c + 10c + c

100a + 10b = 36c.... (2)

Now we know,

a, b, c are natural numbers and hence positive,

So in (2), L.H.S should be positive,

Minimum value of c for making L.H.S positive should be greater than 3.

For c = 4 and a = 1,

100 + 10b = 144

b = 4.4

Again,

For c = 5, a = 1

b = 8

Now,

b = 8 satifies the given condition for natural numbers,

So, we get

a = 1, b = 8 and c = 5

Now,

= (a*b*c)- (a*b) – (a*c) - a - b - c

= 40 - 8 - 5 - 1 - 8 - 5

= 13

Answer 2

a, b, c are singular digits

abc is a 3 digit number.

ccc is a 3 digit number.

abc + abc + abc = ccc.... (1)

We have,

From (1),

3*abc = ccc

We know,

abc and ccc are 3 digit numbers.

A 3 digit number is written like = 100a + 10b + c

So again,

From (1),

3{100a + 10b + c} = 100c + 10c + c

300a + 30b + 3c = 100c + 10c + c

100a + 10b = 36c.... (2)

Now we know,

a, b, c are natural numbers and hence positive,

So in (2), L.H.S should be positive,

Minimum value of c for making L.H.S positive should be greater than 3.

For c = 4 and a = 1,

100 + 10b = 144

b = 4.4

Again,

For c = 5, a = 1

b = 8

Now,

b = 8 satifies the given condition for natural numbers,

So, we get

a = 1, b = 8 and c = 5

Now,

= (a*b*c)- (a*b) – (a*c) - a - b - c

= 40 - 8 - 5 - 1 - 8 - 5

= 13

$\huge\boxed{Thanks}$