Question

how many there digit number are there,those are divided by 9 and 7,but not 4​

Answer 1

Answer:

Let us write the equation in the language of Modular Arithmetic.

Let the 3 digit Number be X

X=0 (mod 9)

X=0 (mod 7)

The above 2 equations can be combined into one equation as

X=0 (mod 63)

1st equation :-

X=0 (mod 63) ———(i)

X=1 (mod 4)————(ii)

2nd equation

X=0 (mod 63)

X=2 (mod 4)

3rd equation

X=0 (mod 63)

X=3 (mod 4)

In case X is not divisible by 4, only 1 Or 2 or 3 may appear as Remainder. So there are 3 possibilities.

Let us solve the 1st. equation

From (i)

X=63 t where t is any integer. (*)

Putting this value in (ii) we get,

63t=1 (mod 4) ———-(iii)

3 is the Multiplicative Inverse of 63 (mod 4)

[ 63=4 (15)+3

4=3 (1)+1

Working back :—

1 =4+3 (-1)

=4+{63+4 (-15)}(-1)

=4+63 (-1)+4 (15)

=4 (16)+63 (-1)

-1 (mod 4)=3 (mod 4) ]

Multiply 3 both the sides of (iii) ,we get

t=3 (mod 4)

t= 4 u +3

Putting the value of t in (*) we get,

X=63 (4u+3)=252 u +189

Take

u=0 ,then X=189

u=1 ,then X=252+189=441

u=2 , X=2*252+189=693

u=3, X=3*252+189=945

u =4, X=1008+189 Discarded as 4 digit number.

—————————————-

Similarly when X=2 (mod 4) 2nd equation

Solutions

X=252 u +126

u=0, X=126

u=1, X=252+126=378

u=2, X=252*2+126=630

u=3, X=252*3+126=882

3rd solution

X=252 u +63

u=0,X=63 Discarded.

u=1, X=315

u=2,X=567

u=3, X=819

So all the Answers are:—

189 , 441 , 693 , 945

378 , 630 ,882

315, , 567 & 819 □ ANSWER.

Hope it helps you.

Happy Studying!! :)