Numbers-IV
This section of numbers chapter mentioned some solved examples, which will helpful for user to understand that how the number system and formulas works:
Ex. 1. Simplify
: (i) 8888 + 888 + 88 + 8
(ii) 11992 -
7823 - 456
Sol. i )
8888 ii)
11992 - 7823 - 456 = 11992 - (7823 + 456)
888
= 11992 - 8279 = 3713
88 7823 11992
+ 8
+ 456 - 8279
9872
8279 3713
Ex. 2: What value will replace the question mark in each of the following equations?
(i) ? -
1936248 = 1635773
(ii) 8597 -
? = 7429 - 4358
Sol.
(i) Let x
- 1936248=1635773.
Then, x = 1635773 + 1936248
x = 3572021.
(ii) Let 8597 - x = 7429 -
4358.
Then, x =
(8597 + 4358) - 7429
= 12955 - 7429
x = 5526.
Ex. 3.
Simplify : (i) 5793405 x 9999 (ii)
839478 x 625
Sol.
i) 5793405 x 9999
= 5793405(10000-1)
= 57934050000-5793405
= 57928256595
ii) 839478 x
625
= 839478 x 54
= 8394780000
16
= 524673750.
Ex. 4. Evaluate : (i) 986 x 237 + 986 x 863 (ii) 983 x 207 - 983 x 107
Sol.
(i) 986 x 137
+ 986 x 863
= 986 x (137 + 863)
= 986 x 1000
= 986000.
(ii) 983 x
207 - 983 x 107
= 983 x (207 - 107)
= 983 x 100
= 98300.
Ex. 5.
Simplify : (i) 1605 x 1605 ii) 1398 x
1398
Sol.
i) 1605 x
1605
= (1605)2
= (1600 + 5)2
= (1600)2 + (5)2
+ 2 x 1600 x 5
= 2560000 + 25 + 16000
= 2576025.
(ii) 1398 x
1398 - (1398)2
= (1400 - 2)2
= (1400)2 + (2)2
- 2 x 1400 x 2
Ex. 6.
Evaluate : (313 x 313 + 287 x 287).
Sol.
(a2 + b2) = 1/2 [(a + b)2
+ (a- b)2]
(313)2
+ (287)2 = 1/2 [(313 + 287)2 + (313 - 287)2]
=
½[(600)2 + (26)2]
= 1/2 (360000
+ 676)
= 180338.
Ex. 7.
Which of the following are prime numbers ?
(i)
241 (ii) 337 (iii) 391 (iv) 571
Sol.
(i) Clearly,
16 > Ö241.
Prime numbers less than 16 are 2, 3, 5, 7, 11, 13.
241 is not divisible by any one of
them.
241 is a prime number.
(ii) Clearly, 19>Ö337.
Prime numbers less than 19 are 2, 3, 5,
7, 11,13,17.
337 is not divisible by any one of
them.
337 is a prime number.
(iii) Clearly, 20 > Ö39l".
Prime numbers less than 20 are 2,
3, 5, 7, 11, 13, 17, 19.
We find that 391 is
divisible by 17.
391 is not prime.
(iv) Clearly, 24 > Ö57T.
Prime numbers less than 24 are 2, 3, 5,
7, 11, 13, 17, 19, 23.
571 is not divisible by
any one of them.
571 is a prime number.
Ex. 8. Find the unit's digit in the product
(2467)163 x (341)72.
Sol.
Clearly, unit's digit in the given product = unit's digit in 7153 x
172.
Now, 74 gives unit digit 1.
7152 gives unit digit 1,
\ 7153
gives unit digit (l x 7) = 7. Also, 172 gives unit digit 1.
Hence, unit's digit in the product = (7
x 1) = 7.
Ex. 09.
Find the unit's digit in (264)102 + (264)103
Sol.
Required unit's digit = unit's digit in (4)102 + (4)103.
Now, 42 gives unit digit 6.
\(4)102 gives unjt digit 6.
\(4)103 gives unit digit of the product (6 x 4) i.e., 4.
Hence, unit's digit in (264)m + (264)103
= unit's digit in (6 + 4) = 0.
Ex. 10. Find the total number of prime factors
in the expression (4)11 x (7)5 x (11)2.
Sol.
(4)11 x (7)5 x (11)2
= (2 x 2)11 x
(7)5 x (11)2
= 211 x 211 x 75 x
112
= 222 x 75 x112
Total number of prime factors = (22 + 5
+ 2) = 29.
Ex.11.
Simplify : (i) 896 x 896 - 204 x 204
(ii) 387 x 387 + 114
x 114 + 2 x 387 x 114
(iii) 81 X 81 +
68 X 68-2 x 81 X 68.
Sol.
(i) Given exp
= (896)2 - (204)2
= (896 + 204) (896 - 204)
= 1100 x 692
= 761200.
(ii) Given
exp
= (387)2+ (114)2+
(2 x 387x 114)
= a2 + b2
+ 2ab, where a = 387,b=114
= (a+b)2
= (387 + 114 )2
= (501)2
= 251001.
(iii) Given
exp
= (81)2 + (68)2 – 2x 81 x 68
= a2 + b2
– 2ab,Where a =81,b=68
= (a-b)2
= (81 –68)2
=
(13)2
=169
Ex.12. Which of the following numbers is divisible by 3 ?
(i) 541326 (ii) 5967013
Sol.
(i) Sum of
digits in 541326 = (5 + 4 + 1 + 3 + 2 + 6) = 21, which is divisible by 3.
Hence, 541326
is divisible by 3.
(ii) Sum of
digits in 5967013 =(5+9 + 6 + 7 + 0+1 +3) = 31, which is not divisible by 3.
Hence,
5967013 is not divisible by 3.
Ex.13.What
least value must be assigned to * so that the number 197*5462 is r 9 ?
Sol.
Let the
missing digit be x.
Sum of digits
= (1 + 9 + 7 + x + 5 + 4 + 6 +2) = (34 + x).
For (34 + x)
to be divisible by 9, x must be replaced by 2 .
Hence, the
digit in place of * must be 2.
Ex. 14.
Which of the following numbers is divisible by 4 ?
(i)
67920594 (ii) 618703572
Sol.
(i) The
number formed by the last two digits in the given number is 94, which is not
divisible by 4.
Hence,
67920594 is not divisible by 4.
(ii) The
number formed by the last two digits in the given number is 72, which is
divisible by 4.
Hence,
618703572 is divisible by 4.
Ex.30.How
many terms are there in 2,4,8,16……1024?
Sol.
Clearly
2,4,8,16……..1024 form a GP. With a=2 and r = 4/2 =2.
Let the number of terms be n .
Then 2 x 2n-1 =1024 or 2n-1
=512 = 29.
\n-1=9 or n=10.
Note: In next post we will start H.C.F and L.C.M of Numbers.