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CBSE Class 10  >  Class 10 Notes  >  10_math_imp_ch1_3

10_math_imp_ch1_3

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 Page 1


M a t e r i a l 	 d o w n l o a d e d 	 f r o m 	 m y C B S E g u i d e . c o m .
1 	 / 	 8
C B S E 	 C l a s s 	 1 0 	 M a t h e m a t i c s
I m p o r t a n t 	 Q u e s t i o n s
C h a p t e r 	 1
R e a l 	 N u m b e r s
3 	 M a r k s 	 Q u e s t i o n s
1 . 	 U s e 	 E u c l i d ’ s 	 d i v i s i o n 	 a l g o r i t h m 	 t o 	 f i n d 	 t h e 	 H C F 	 o f :
( i ) 	 1 3 5 	 a n d 	 2 2 5
( i i ) 	 1 9 6 	 a n d 	 3 8 2 2 0
( i i i ) 	 8 6 7 	 a n d 	 2 5 5
A n s . 	 ( i ) 	 1 3 5 	 a n d 	 2 2 5
W e 	 h a v e 	 2 2 5 	 > 	 1 3 5 ,
S o , 	 w e 	 a p p l y 	 t h e 	 d i v i s i o n 	 l e m m a 	 t o 	 2 2 5 	 a n d 	 1 3 5 	 t o 	 o b t a i n
2 2 5 	 = 	 1 3 5 	 × 	 1 	 + 	 9 0
H e r e 	 r e m a i n d e r 	 9 0 	 ? 	 0 , 	 w e 	 a p p l y 	 t h e 	 d i v i s i o n 	 l e m m a 	 a g a i n 	 t o 	 1 3 5 	 a n d 	 9 0 	 t o 	 o b t a i n
1 3 5 	 = 	 9 0 	 × 	 1 	 + 	 4 5
W e 	 c o n s i d e r 	 t h e 	 n e w 	 d i v i s o r 	 9 0 	 a n d 	 n e w 	 r e m a i n d e r 	 4 5 ? 	 0 , 	 a n d 	 a p p l y 	 t h e 	 d i v i s i o n 	 l e m m a 	 t o
o b t a i n
9 0 	 = 	 2 	 × 	 4 5 	 + 	 0
S i n c e 	 t h a t 	 t i m e 	 t h e 	 r e m a i n d e r 	 i s 	 z e r o , 	 t h e 	 p r o c e s s 	 g e t 	 s t o p s .
T h e 	 d i v i s o r 	 a t 	 t h i s 	 s t a g e 	 i s 	 4 5
T h e r e f o r e , 	 t h e 	 H C F 	 o f 	 1 3 5 	 a n d 	 2 2 5 	 i s 	 4 5 .
( i i ) 	 1 9 6 	 a n d 	 3 8 2 2 0
Page 2


M a t e r i a l 	 d o w n l o a d e d 	 f r o m 	 m y C B S E g u i d e . c o m .
1 	 / 	 8
C B S E 	 C l a s s 	 1 0 	 M a t h e m a t i c s
I m p o r t a n t 	 Q u e s t i o n s
C h a p t e r 	 1
R e a l 	 N u m b e r s
3 	 M a r k s 	 Q u e s t i o n s
1 . 	 U s e 	 E u c l i d ’ s 	 d i v i s i o n 	 a l g o r i t h m 	 t o 	 f i n d 	 t h e 	 H C F 	 o f :
( i ) 	 1 3 5 	 a n d 	 2 2 5
( i i ) 	 1 9 6 	 a n d 	 3 8 2 2 0
( i i i ) 	 8 6 7 	 a n d 	 2 5 5
A n s . 	 ( i ) 	 1 3 5 	 a n d 	 2 2 5
W e 	 h a v e 	 2 2 5 	 > 	 1 3 5 ,
S o , 	 w e 	 a p p l y 	 t h e 	 d i v i s i o n 	 l e m m a 	 t o 	 2 2 5 	 a n d 	 1 3 5 	 t o 	 o b t a i n
2 2 5 	 = 	 1 3 5 	 × 	 1 	 + 	 9 0
H e r e 	 r e m a i n d e r 	 9 0 	 ? 	 0 , 	 w e 	 a p p l y 	 t h e 	 d i v i s i o n 	 l e m m a 	 a g a i n 	 t o 	 1 3 5 	 a n d 	 9 0 	 t o 	 o b t a i n
1 3 5 	 = 	 9 0 	 × 	 1 	 + 	 4 5
W e 	 c o n s i d e r 	 t h e 	 n e w 	 d i v i s o r 	 9 0 	 a n d 	 n e w 	 r e m a i n d e r 	 4 5 ? 	 0 , 	 a n d 	 a p p l y 	 t h e 	 d i v i s i o n 	 l e m m a 	 t o
o b t a i n
9 0 	 = 	 2 	 × 	 4 5 	 + 	 0
S i n c e 	 t h a t 	 t i m e 	 t h e 	 r e m a i n d e r 	 i s 	 z e r o , 	 t h e 	 p r o c e s s 	 g e t 	 s t o p s .
T h e 	 d i v i s o r 	 a t 	 t h i s 	 s t a g e 	 i s 	 4 5
T h e r e f o r e , 	 t h e 	 H C F 	 o f 	 1 3 5 	 a n d 	 2 2 5 	 i s 	 4 5 .
( i i ) 	 1 9 6 	 a n d 	 3 8 2 2 0
M a t e r i a l 	 d o w n l o a d e d 	 f r o m 	 m y C B S E g u i d e . c o m .
2 	 / 	 8
W e 	 h a v e 	 3 8 2 2 0 	 > 	 1 9 6 ,
S o , 	 w e 	 a p p l y 	 t h e 	 d i v i s i o n 	 l e m m a 	 t o 	 3 8 2 2 0 	 a n d 	 1 9 6 	 t o 	 o b t a i n
3 8 2 2 0 	 = 	 1 9 6 	 × 	 1 9 5 	 + 	 0
S i n c e 	 w e 	 g e t 	 t h e 	 r e m a i n d e r 	 i s 	 z e r o , 	 t h e 	 p r o c e s s 	 s t o p s .
T h e 	 d i v i s o r 	 a t 	 t h i s 	 s t a g e 	 i s 	 1 9 6 ,
T h e r e f o r e , 	 H C F 	 o f 	 1 9 6 	 a n d 	 3 8 2 2 0 	 i s 	 1 9 6 .
( i i i ) 	 8 6 7 	 a n d 	 2 5 5
W e 	 h a v e 	 8 6 7 	 > 	 2 5 5 ,
S o , 	 w e 	 a p p l y 	 t h e 	 d i v i s i o n 	 l e m m a 	 t o 	 8 6 7 	 a n d 	 2 5 5 	 t o 	 o b t a i n
8 6 7 	 = 	 2 5 5 	 × 	 3 	 + 	 1 0 2
H e r e 	 r e m a i n d e r 	 1 0 2 	 ? 	 0 , 	 w e 	 a p p l y 	 t h e 	 d i v i s i o n 	 l e m m a 	 a g a i n 	 t o 	 2 5 5 	 a n d 	 1 0 2 	 t o 	 o b t a i n
2 5 5 	 = 	 1 0 2 	 × 	 2 	 + 	 5 1
H e r e 	 r e m a i n d e r 	 5 1 	 ? 	 0 , 	 w e 	 a p p l y 	 t h e 	 d i v i s i o n 	 l e m m a 	 a g a i n 	 t o 	 1 0 2 	 a n d 	 5 1 	 t o 	 o b t a i n
1 0 2 	 = 	 5 1 	 × 	 2 	 + 	 0
S i n c e 	 w e 	 g e t 	 t h e 	 r e m a i n d e r 	 i s 	 z e r o , 	 t h e 	 p r o c e s s 	 s t o p s .
T h e 	 d i v i s o r 	 a t 	 t h i s 	 s t a g e 	 i s 	 5 1 ,
T h e r e f o r e , 	 H C F 	 o f 	 8 6 7 	 a n d 	 2 5 5 	 i s 	 5 1 .
2 . 	 F i n d 	 t h e 	 g r e a t e s t 	 n u m b e r 	 o f 	 6 	 d i g i t s 	 e x a c t l y 	 d i v i s i b l e 	 b y 	 2 4 , 	 1 5 	 a n d 	 3 6 .
A n s . 	 T h e 	 g r e a t e r 	 n u m b e r 	 o f 	 6 	 d i g i t s 	 i s 	 9 9 9 9 9 9 .
L C M 	 o f 	 2 4 , 	 1 5 , 	 a n d 	 3 6 	 i s 	 3 6 0 .
Page 3


M a t e r i a l 	 d o w n l o a d e d 	 f r o m 	 m y C B S E g u i d e . c o m .
1 	 / 	 8
C B S E 	 C l a s s 	 1 0 	 M a t h e m a t i c s
I m p o r t a n t 	 Q u e s t i o n s
C h a p t e r 	 1
R e a l 	 N u m b e r s
3 	 M a r k s 	 Q u e s t i o n s
1 . 	 U s e 	 E u c l i d ’ s 	 d i v i s i o n 	 a l g o r i t h m 	 t o 	 f i n d 	 t h e 	 H C F 	 o f :
( i ) 	 1 3 5 	 a n d 	 2 2 5
( i i ) 	 1 9 6 	 a n d 	 3 8 2 2 0
( i i i ) 	 8 6 7 	 a n d 	 2 5 5
A n s . 	 ( i ) 	 1 3 5 	 a n d 	 2 2 5
W e 	 h a v e 	 2 2 5 	 > 	 1 3 5 ,
S o , 	 w e 	 a p p l y 	 t h e 	 d i v i s i o n 	 l e m m a 	 t o 	 2 2 5 	 a n d 	 1 3 5 	 t o 	 o b t a i n
2 2 5 	 = 	 1 3 5 	 × 	 1 	 + 	 9 0
H e r e 	 r e m a i n d e r 	 9 0 	 ? 	 0 , 	 w e 	 a p p l y 	 t h e 	 d i v i s i o n 	 l e m m a 	 a g a i n 	 t o 	 1 3 5 	 a n d 	 9 0 	 t o 	 o b t a i n
1 3 5 	 = 	 9 0 	 × 	 1 	 + 	 4 5
W e 	 c o n s i d e r 	 t h e 	 n e w 	 d i v i s o r 	 9 0 	 a n d 	 n e w 	 r e m a i n d e r 	 4 5 ? 	 0 , 	 a n d 	 a p p l y 	 t h e 	 d i v i s i o n 	 l e m m a 	 t o
o b t a i n
9 0 	 = 	 2 	 × 	 4 5 	 + 	 0
S i n c e 	 t h a t 	 t i m e 	 t h e 	 r e m a i n d e r 	 i s 	 z e r o , 	 t h e 	 p r o c e s s 	 g e t 	 s t o p s .
T h e 	 d i v i s o r 	 a t 	 t h i s 	 s t a g e 	 i s 	 4 5
T h e r e f o r e , 	 t h e 	 H C F 	 o f 	 1 3 5 	 a n d 	 2 2 5 	 i s 	 4 5 .
( i i ) 	 1 9 6 	 a n d 	 3 8 2 2 0
M a t e r i a l 	 d o w n l o a d e d 	 f r o m 	 m y C B S E g u i d e . c o m .
2 	 / 	 8
W e 	 h a v e 	 3 8 2 2 0 	 > 	 1 9 6 ,
S o , 	 w e 	 a p p l y 	 t h e 	 d i v i s i o n 	 l e m m a 	 t o 	 3 8 2 2 0 	 a n d 	 1 9 6 	 t o 	 o b t a i n
3 8 2 2 0 	 = 	 1 9 6 	 × 	 1 9 5 	 + 	 0
S i n c e 	 w e 	 g e t 	 t h e 	 r e m a i n d e r 	 i s 	 z e r o , 	 t h e 	 p r o c e s s 	 s t o p s .
T h e 	 d i v i s o r 	 a t 	 t h i s 	 s t a g e 	 i s 	 1 9 6 ,
T h e r e f o r e , 	 H C F 	 o f 	 1 9 6 	 a n d 	 3 8 2 2 0 	 i s 	 1 9 6 .
( i i i ) 	 8 6 7 	 a n d 	 2 5 5
W e 	 h a v e 	 8 6 7 	 > 	 2 5 5 ,
S o , 	 w e 	 a p p l y 	 t h e 	 d i v i s i o n 	 l e m m a 	 t o 	 8 6 7 	 a n d 	 2 5 5 	 t o 	 o b t a i n
8 6 7 	 = 	 2 5 5 	 × 	 3 	 + 	 1 0 2
H e r e 	 r e m a i n d e r 	 1 0 2 	 ? 	 0 , 	 w e 	 a p p l y 	 t h e 	 d i v i s i o n 	 l e m m a 	 a g a i n 	 t o 	 2 5 5 	 a n d 	 1 0 2 	 t o 	 o b t a i n
2 5 5 	 = 	 1 0 2 	 × 	 2 	 + 	 5 1
H e r e 	 r e m a i n d e r 	 5 1 	 ? 	 0 , 	 w e 	 a p p l y 	 t h e 	 d i v i s i o n 	 l e m m a 	 a g a i n 	 t o 	 1 0 2 	 a n d 	 5 1 	 t o 	 o b t a i n
1 0 2 	 = 	 5 1 	 × 	 2 	 + 	 0
S i n c e 	 w e 	 g e t 	 t h e 	 r e m a i n d e r 	 i s 	 z e r o , 	 t h e 	 p r o c e s s 	 s t o p s .
T h e 	 d i v i s o r 	 a t 	 t h i s 	 s t a g e 	 i s 	 5 1 ,
T h e r e f o r e , 	 H C F 	 o f 	 8 6 7 	 a n d 	 2 5 5 	 i s 	 5 1 .
2 . 	 F i n d 	 t h e 	 g r e a t e s t 	 n u m b e r 	 o f 	 6 	 d i g i t s 	 e x a c t l y 	 d i v i s i b l e 	 b y 	 2 4 , 	 1 5 	 a n d 	 3 6 .
A n s . 	 T h e 	 g r e a t e r 	 n u m b e r 	 o f 	 6 	 d i g i t s 	 i s 	 9 9 9 9 9 9 .
L C M 	 o f 	 2 4 , 	 1 5 , 	 a n d 	 3 6 	 i s 	 3 6 0 .
M a t e r i a l 	 d o w n l o a d e d 	 f r o m 	 m y C B S E g u i d e . c o m .
3 	 / 	 8
3 . 	 P r o v e 	 t h a t 	 t h e 	 s q u a r e 	 o f 	 a n y 	 p o s i t i v e 	 i n t e g e r 	 i s 	 o f 	 t h e 	 f o r m 	 4 q 	 o r 	 4 q 	 + 	 1 	 f o r 	 s o m e
i n t e g e r 	 q .
A n s . 	 L e t 	 a 	 = 	 4 q 	 + 	 r , 	 w h e n 	 r 	 = 	 0 , 	 1 , 	 2 	 a n d 	 3
	 	 N u m b e r s 	 a r e 	 4 q , 	 4 q 	 + 	 1 , 	 4 q 	 + 	 2 	 a n d 	 4 q 	 + 	 3
4 . 	 1 4 4 	 c a r t o n s 	 o f 	 c o k e 	 c a n 	 a n d 	 9 0 	 c a r t o n s 	 o f 	 P e p s i 	 c a n 	 a r e 	 t o 	 b e 	 s t a c k e d 	 i n 	 a 	 c a n t e e n . 	 I f
e a c h 	 s t a c k 	 i s 	 o f 	 t h e 	 s a m e 	 h e i g h t 	 a n d 	 i s 	 t o 	 c o n t a i n 	 c a r t o n s 	 o f 	 t h e 	 s a m e 	 d r i n k . 	 W h a t
w o u l d 	 b e 	 t h e 	 g r e a t e r 	 n u m b e r 	 o f 	 c a r t o n s 	 e a c h 	 s t a c k 	 w o u l d 	 h a v e ?
A n s . 	 W e 	 f i n d 	 t h e 	 H C F 	 o f 	 1 4 4 	 a n d 	 9 0
5 . 	 P r o v e 	 t h a t 	 p r o d u c t 	 o f 	 t h r e e 	 c o n s e c u t i v e 	 p o s i t i v e 	 i n t e g e r s 	 i s 	 d i v i s i b l e 	 b y 	 6 . 	 	
A n s . 	 L e t 	 t h r e e 	 c o n s e c u t i v e 	 n u m b e r s 	 b e 	 x , 	 ( x 	 + 	 1 ) 	 a n d 	 ( x 	 + 	 2 )
L e t 	 x 	 = 	 6 q 	 + 	 r ; 	 0
Page 4


M a t e r i a l 	 d o w n l o a d e d 	 f r o m 	 m y C B S E g u i d e . c o m .
1 	 / 	 8
C B S E 	 C l a s s 	 1 0 	 M a t h e m a t i c s
I m p o r t a n t 	 Q u e s t i o n s
C h a p t e r 	 1
R e a l 	 N u m b e r s
3 	 M a r k s 	 Q u e s t i o n s
1 . 	 U s e 	 E u c l i d ’ s 	 d i v i s i o n 	 a l g o r i t h m 	 t o 	 f i n d 	 t h e 	 H C F 	 o f :
( i ) 	 1 3 5 	 a n d 	 2 2 5
( i i ) 	 1 9 6 	 a n d 	 3 8 2 2 0
( i i i ) 	 8 6 7 	 a n d 	 2 5 5
A n s . 	 ( i ) 	 1 3 5 	 a n d 	 2 2 5
W e 	 h a v e 	 2 2 5 	 > 	 1 3 5 ,
S o , 	 w e 	 a p p l y 	 t h e 	 d i v i s i o n 	 l e m m a 	 t o 	 2 2 5 	 a n d 	 1 3 5 	 t o 	 o b t a i n
2 2 5 	 = 	 1 3 5 	 × 	 1 	 + 	 9 0
H e r e 	 r e m a i n d e r 	 9 0 	 ? 	 0 , 	 w e 	 a p p l y 	 t h e 	 d i v i s i o n 	 l e m m a 	 a g a i n 	 t o 	 1 3 5 	 a n d 	 9 0 	 t o 	 o b t a i n
1 3 5 	 = 	 9 0 	 × 	 1 	 + 	 4 5
W e 	 c o n s i d e r 	 t h e 	 n e w 	 d i v i s o r 	 9 0 	 a n d 	 n e w 	 r e m a i n d e r 	 4 5 ? 	 0 , 	 a n d 	 a p p l y 	 t h e 	 d i v i s i o n 	 l e m m a 	 t o
o b t a i n
9 0 	 = 	 2 	 × 	 4 5 	 + 	 0
S i n c e 	 t h a t 	 t i m e 	 t h e 	 r e m a i n d e r 	 i s 	 z e r o , 	 t h e 	 p r o c e s s 	 g e t 	 s t o p s .
T h e 	 d i v i s o r 	 a t 	 t h i s 	 s t a g e 	 i s 	 4 5
T h e r e f o r e , 	 t h e 	 H C F 	 o f 	 1 3 5 	 a n d 	 2 2 5 	 i s 	 4 5 .
( i i ) 	 1 9 6 	 a n d 	 3 8 2 2 0
M a t e r i a l 	 d o w n l o a d e d 	 f r o m 	 m y C B S E g u i d e . c o m .
2 	 / 	 8
W e 	 h a v e 	 3 8 2 2 0 	 > 	 1 9 6 ,
S o , 	 w e 	 a p p l y 	 t h e 	 d i v i s i o n 	 l e m m a 	 t o 	 3 8 2 2 0 	 a n d 	 1 9 6 	 t o 	 o b t a i n
3 8 2 2 0 	 = 	 1 9 6 	 × 	 1 9 5 	 + 	 0
S i n c e 	 w e 	 g e t 	 t h e 	 r e m a i n d e r 	 i s 	 z e r o , 	 t h e 	 p r o c e s s 	 s t o p s .
T h e 	 d i v i s o r 	 a t 	 t h i s 	 s t a g e 	 i s 	 1 9 6 ,
T h e r e f o r e , 	 H C F 	 o f 	 1 9 6 	 a n d 	 3 8 2 2 0 	 i s 	 1 9 6 .
( i i i ) 	 8 6 7 	 a n d 	 2 5 5
W e 	 h a v e 	 8 6 7 	 > 	 2 5 5 ,
S o , 	 w e 	 a p p l y 	 t h e 	 d i v i s i o n 	 l e m m a 	 t o 	 8 6 7 	 a n d 	 2 5 5 	 t o 	 o b t a i n
8 6 7 	 = 	 2 5 5 	 × 	 3 	 + 	 1 0 2
H e r e 	 r e m a i n d e r 	 1 0 2 	 ? 	 0 , 	 w e 	 a p p l y 	 t h e 	 d i v i s i o n 	 l e m m a 	 a g a i n 	 t o 	 2 5 5 	 a n d 	 1 0 2 	 t o 	 o b t a i n
2 5 5 	 = 	 1 0 2 	 × 	 2 	 + 	 5 1
H e r e 	 r e m a i n d e r 	 5 1 	 ? 	 0 , 	 w e 	 a p p l y 	 t h e 	 d i v i s i o n 	 l e m m a 	 a g a i n 	 t o 	 1 0 2 	 a n d 	 5 1 	 t o 	 o b t a i n
1 0 2 	 = 	 5 1 	 × 	 2 	 + 	 0
S i n c e 	 w e 	 g e t 	 t h e 	 r e m a i n d e r 	 i s 	 z e r o , 	 t h e 	 p r o c e s s 	 s t o p s .
T h e 	 d i v i s o r 	 a t 	 t h i s 	 s t a g e 	 i s 	 5 1 ,
T h e r e f o r e , 	 H C F 	 o f 	 8 6 7 	 a n d 	 2 5 5 	 i s 	 5 1 .
2 . 	 F i n d 	 t h e 	 g r e a t e s t 	 n u m b e r 	 o f 	 6 	 d i g i t s 	 e x a c t l y 	 d i v i s i b l e 	 b y 	 2 4 , 	 1 5 	 a n d 	 3 6 .
A n s . 	 T h e 	 g r e a t e r 	 n u m b e r 	 o f 	 6 	 d i g i t s 	 i s 	 9 9 9 9 9 9 .
L C M 	 o f 	 2 4 , 	 1 5 , 	 a n d 	 3 6 	 i s 	 3 6 0 .
M a t e r i a l 	 d o w n l o a d e d 	 f r o m 	 m y C B S E g u i d e . c o m .
3 	 / 	 8
3 . 	 P r o v e 	 t h a t 	 t h e 	 s q u a r e 	 o f 	 a n y 	 p o s i t i v e 	 i n t e g e r 	 i s 	 o f 	 t h e 	 f o r m 	 4 q 	 o r 	 4 q 	 + 	 1 	 f o r 	 s o m e
i n t e g e r 	 q .
A n s . 	 L e t 	 a 	 = 	 4 q 	 + 	 r , 	 w h e n 	 r 	 = 	 0 , 	 1 , 	 2 	 a n d 	 3
	 	 N u m b e r s 	 a r e 	 4 q , 	 4 q 	 + 	 1 , 	 4 q 	 + 	 2 	 a n d 	 4 q 	 + 	 3
4 . 	 1 4 4 	 c a r t o n s 	 o f 	 c o k e 	 c a n 	 a n d 	 9 0 	 c a r t o n s 	 o f 	 P e p s i 	 c a n 	 a r e 	 t o 	 b e 	 s t a c k e d 	 i n 	 a 	 c a n t e e n . 	 I f
e a c h 	 s t a c k 	 i s 	 o f 	 t h e 	 s a m e 	 h e i g h t 	 a n d 	 i s 	 t o 	 c o n t a i n 	 c a r t o n s 	 o f 	 t h e 	 s a m e 	 d r i n k . 	 W h a t
w o u l d 	 b e 	 t h e 	 g r e a t e r 	 n u m b e r 	 o f 	 c a r t o n s 	 e a c h 	 s t a c k 	 w o u l d 	 h a v e ?
A n s . 	 W e 	 f i n d 	 t h e 	 H C F 	 o f 	 1 4 4 	 a n d 	 9 0
5 . 	 P r o v e 	 t h a t 	 p r o d u c t 	 o f 	 t h r e e 	 c o n s e c u t i v e 	 p o s i t i v e 	 i n t e g e r s 	 i s 	 d i v i s i b l e 	 b y 	 6 . 	 	
A n s . 	 L e t 	 t h r e e 	 c o n s e c u t i v e 	 n u m b e r s 	 b e 	 x , 	 ( x 	 + 	 1 ) 	 a n d 	 ( x 	 + 	 2 )
L e t 	 x 	 = 	 6 q 	 + 	 r ; 	 0
M a t e r i a l 	 d o w n l o a d e d 	 f r o m 	 m y C B S E g u i d e . c o m .
4 	 / 	 8
6 . 	 P r o v e 	 t h a t 	 ( 3 	 – ) 	 i s 	 a n 	 i r r a t i o n a l 	 n u m b e r .
Page 5


M a t e r i a l 	 d o w n l o a d e d 	 f r o m 	 m y C B S E g u i d e . c o m .
1 	 / 	 8
C B S E 	 C l a s s 	 1 0 	 M a t h e m a t i c s
I m p o r t a n t 	 Q u e s t i o n s
C h a p t e r 	 1
R e a l 	 N u m b e r s
3 	 M a r k s 	 Q u e s t i o n s
1 . 	 U s e 	 E u c l i d ’ s 	 d i v i s i o n 	 a l g o r i t h m 	 t o 	 f i n d 	 t h e 	 H C F 	 o f :
( i ) 	 1 3 5 	 a n d 	 2 2 5
( i i ) 	 1 9 6 	 a n d 	 3 8 2 2 0
( i i i ) 	 8 6 7 	 a n d 	 2 5 5
A n s . 	 ( i ) 	 1 3 5 	 a n d 	 2 2 5
W e 	 h a v e 	 2 2 5 	 > 	 1 3 5 ,
S o , 	 w e 	 a p p l y 	 t h e 	 d i v i s i o n 	 l e m m a 	 t o 	 2 2 5 	 a n d 	 1 3 5 	 t o 	 o b t a i n
2 2 5 	 = 	 1 3 5 	 × 	 1 	 + 	 9 0
H e r e 	 r e m a i n d e r 	 9 0 	 ? 	 0 , 	 w e 	 a p p l y 	 t h e 	 d i v i s i o n 	 l e m m a 	 a g a i n 	 t o 	 1 3 5 	 a n d 	 9 0 	 t o 	 o b t a i n
1 3 5 	 = 	 9 0 	 × 	 1 	 + 	 4 5
W e 	 c o n s i d e r 	 t h e 	 n e w 	 d i v i s o r 	 9 0 	 a n d 	 n e w 	 r e m a i n d e r 	 4 5 ? 	 0 , 	 a n d 	 a p p l y 	 t h e 	 d i v i s i o n 	 l e m m a 	 t o
o b t a i n
9 0 	 = 	 2 	 × 	 4 5 	 + 	 0
S i n c e 	 t h a t 	 t i m e 	 t h e 	 r e m a i n d e r 	 i s 	 z e r o , 	 t h e 	 p r o c e s s 	 g e t 	 s t o p s .
T h e 	 d i v i s o r 	 a t 	 t h i s 	 s t a g e 	 i s 	 4 5
T h e r e f o r e , 	 t h e 	 H C F 	 o f 	 1 3 5 	 a n d 	 2 2 5 	 i s 	 4 5 .
( i i ) 	 1 9 6 	 a n d 	 3 8 2 2 0
M a t e r i a l 	 d o w n l o a d e d 	 f r o m 	 m y C B S E g u i d e . c o m .
2 	 / 	 8
W e 	 h a v e 	 3 8 2 2 0 	 > 	 1 9 6 ,
S o , 	 w e 	 a p p l y 	 t h e 	 d i v i s i o n 	 l e m m a 	 t o 	 3 8 2 2 0 	 a n d 	 1 9 6 	 t o 	 o b t a i n
3 8 2 2 0 	 = 	 1 9 6 	 × 	 1 9 5 	 + 	 0
S i n c e 	 w e 	 g e t 	 t h e 	 r e m a i n d e r 	 i s 	 z e r o , 	 t h e 	 p r o c e s s 	 s t o p s .
T h e 	 d i v i s o r 	 a t 	 t h i s 	 s t a g e 	 i s 	 1 9 6 ,
T h e r e f o r e , 	 H C F 	 o f 	 1 9 6 	 a n d 	 3 8 2 2 0 	 i s 	 1 9 6 .
( i i i ) 	 8 6 7 	 a n d 	 2 5 5
W e 	 h a v e 	 8 6 7 	 > 	 2 5 5 ,
S o , 	 w e 	 a p p l y 	 t h e 	 d i v i s i o n 	 l e m m a 	 t o 	 8 6 7 	 a n d 	 2 5 5 	 t o 	 o b t a i n
8 6 7 	 = 	 2 5 5 	 × 	 3 	 + 	 1 0 2
H e r e 	 r e m a i n d e r 	 1 0 2 	 ? 	 0 , 	 w e 	 a p p l y 	 t h e 	 d i v i s i o n 	 l e m m a 	 a g a i n 	 t o 	 2 5 5 	 a n d 	 1 0 2 	 t o 	 o b t a i n
2 5 5 	 = 	 1 0 2 	 × 	 2 	 + 	 5 1
H e r e 	 r e m a i n d e r 	 5 1 	 ? 	 0 , 	 w e 	 a p p l y 	 t h e 	 d i v i s i o n 	 l e m m a 	 a g a i n 	 t o 	 1 0 2 	 a n d 	 5 1 	 t o 	 o b t a i n
1 0 2 	 = 	 5 1 	 × 	 2 	 + 	 0
S i n c e 	 w e 	 g e t 	 t h e 	 r e m a i n d e r 	 i s 	 z e r o , 	 t h e 	 p r o c e s s 	 s t o p s .
T h e 	 d i v i s o r 	 a t 	 t h i s 	 s t a g e 	 i s 	 5 1 ,
T h e r e f o r e , 	 H C F 	 o f 	 8 6 7 	 a n d 	 2 5 5 	 i s 	 5 1 .
2 . 	 F i n d 	 t h e 	 g r e a t e s t 	 n u m b e r 	 o f 	 6 	 d i g i t s 	 e x a c t l y 	 d i v i s i b l e 	 b y 	 2 4 , 	 1 5 	 a n d 	 3 6 .
A n s . 	 T h e 	 g r e a t e r 	 n u m b e r 	 o f 	 6 	 d i g i t s 	 i s 	 9 9 9 9 9 9 .
L C M 	 o f 	 2 4 , 	 1 5 , 	 a n d 	 3 6 	 i s 	 3 6 0 .
M a t e r i a l 	 d o w n l o a d e d 	 f r o m 	 m y C B S E g u i d e . c o m .
3 	 / 	 8
3 . 	 P r o v e 	 t h a t 	 t h e 	 s q u a r e 	 o f 	 a n y 	 p o s i t i v e 	 i n t e g e r 	 i s 	 o f 	 t h e 	 f o r m 	 4 q 	 o r 	 4 q 	 + 	 1 	 f o r 	 s o m e
i n t e g e r 	 q .
A n s . 	 L e t 	 a 	 = 	 4 q 	 + 	 r , 	 w h e n 	 r 	 = 	 0 , 	 1 , 	 2 	 a n d 	 3
	 	 N u m b e r s 	 a r e 	 4 q , 	 4 q 	 + 	 1 , 	 4 q 	 + 	 2 	 a n d 	 4 q 	 + 	 3
4 . 	 1 4 4 	 c a r t o n s 	 o f 	 c o k e 	 c a n 	 a n d 	 9 0 	 c a r t o n s 	 o f 	 P e p s i 	 c a n 	 a r e 	 t o 	 b e 	 s t a c k e d 	 i n 	 a 	 c a n t e e n . 	 I f
e a c h 	 s t a c k 	 i s 	 o f 	 t h e 	 s a m e 	 h e i g h t 	 a n d 	 i s 	 t o 	 c o n t a i n 	 c a r t o n s 	 o f 	 t h e 	 s a m e 	 d r i n k . 	 W h a t
w o u l d 	 b e 	 t h e 	 g r e a t e r 	 n u m b e r 	 o f 	 c a r t o n s 	 e a c h 	 s t a c k 	 w o u l d 	 h a v e ?
A n s . 	 W e 	 f i n d 	 t h e 	 H C F 	 o f 	 1 4 4 	 a n d 	 9 0
5 . 	 P r o v e 	 t h a t 	 p r o d u c t 	 o f 	 t h r e e 	 c o n s e c u t i v e 	 p o s i t i v e 	 i n t e g e r s 	 i s 	 d i v i s i b l e 	 b y 	 6 . 	 	
A n s . 	 L e t 	 t h r e e 	 c o n s e c u t i v e 	 n u m b e r s 	 b e 	 x , 	 ( x 	 + 	 1 ) 	 a n d 	 ( x 	 + 	 2 )
L e t 	 x 	 = 	 6 q 	 + 	 r ; 	 0
M a t e r i a l 	 d o w n l o a d e d 	 f r o m 	 m y C B S E g u i d e . c o m .
4 	 / 	 8
6 . 	 P r o v e 	 t h a t 	 ( 3 	 – ) 	 i s 	 a n 	 i r r a t i o n a l 	 n u m b e r .
M a t e r i a l 	 d o w n l o a d e d 	 f r o m 	 m y C B S E g u i d e . c o m .
5 	 / 	 8
A n s . 	
7 . 	 P r o v e 	 t h a t 	 i f 	 x 	 a n d 	 y 	 a r e 	 o d d 	 p o s i t i v e 	 i n t e g e r s , 	 t h e n 	 x
2 	
+ 	 y
2
	 i s 	 e v e n 	 b u t 	 n o t 	 d i v i s i b l e 	 b y
4 .
A n s . 	 L e t 	 x 	 = 	 2 p 	 + 	 1 	 a n d 	 y 	 = 	 2 q 	 + 	 1
8 . 	 S h o w 	 t h a t 	 o n e 	 a n d 	 o n l y 	 o n e 	 o u t 	 o f 	 n , 	 ( n 	 + 	 2 ) 	 o r 	 ( n 	 + 	 4 ) 	 i s 	 d i v i s i b l e 	 b y 	 3 , 	 w h e r e 	 n N . 	 	
A n s . 	 L e t 	 t h e 	 n u m b e r 	 b e 	 ( 3 q 	 + 	 r )
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Apr 19, 2026 Last updated
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