1. When the positive integers a, b and c are divided by 13, the respective remainders are 9, 7 and 10. When a + 2b + 3c is divided by 13, what is the remainder 

Asked by priyankajaju | 28th Jul, 2016, 06:26: PM

Expert Answer:

begin mathsize 20px style G i v e n space t h r e e space n u m b e r s space a r e space a space comma b space a n d space c.
N o w space a s space g i v e n space w h e n space t h e s e space a r e space d i v i d e d space b y space 13 space t h e y space g i v e space r e m a i n d e r space 9 comma 7 space a n d space 10 space r e s p e c t i v e l y.
T h e r e f o r e space b y space E u c l i d space D i v i s i o n space L e m m a space f o r space p o s i t i v e space n u m b e r s space a comma b space a n d space c space t h e r e space e x i s t space u n i q u e space i n t e g e r s
x space comma y space a n d space z space s u c h space t h a t colon

a equals 13 x plus 9
b equals 13 y plus 7
c equals 13 z plus 10
n o w space f i n d i n g space t h e space v a l u e space o f space a plus 2 b plus 3 c space g i v e s space
a plus 2 b plus 3 c equals 13 x plus 9 plus 26 y plus 14 plus 39 z plus 30
a plus 2 b plus 3 c equals 13 x plus 26 y plus 39 z plus 53
W h e n space d i v i d e d space b y space 13 space g i v e s space r e m a i n d e r space a s space 1
A s space w e space c a n space s e e space t h a t space 13 x space 26 y space a n d space 39 z space a r e space d i v i s i b l e space b y space 13 space l e a v i n g space n o space r e m a i n d e r space a s space t h e y space a r e space m u l t i p l e space o f space 13 space i t s e l f
b u t space w h e n space 53 space i s space d i v i d e d space b y space 13 space i t space g i v e s space 1 space a s space r e m a i n d e r
h e n c e comma space a plus 2 b plus 3 c space d i v i d e d space b y space 13 space g i v e s space 1 space a s space r e m a i n d e r. space

end style

Answered by Vijaykumar Wani | 29th Jul, 2016, 11:53: AM