Number system

Question 1

Cyclicity of digits

Answer 1

2-2,4,8,6
3-3,9,7,1
4-4,6
5,6-only nos,
7-7,9,3,1
8-8,4,2,6
9-9,1,

Question 2

Prime nos

Answer 2

1-100=25 prims nos
1-200,=46 prime nos
201-400=32 prime nos
401-600=31 pirm nos
601-800=30 prime nis
PRIMES NOS =6K+/-1 which are not are multiples of 5 and 7
801-1000=29

Question 3

FIND TOTAL NOS FROM A-B
BETWEEN A-B

Answer 3

B-A+1
(B-A-1)-2

Question 4

REMAINDER THERORM EG

Answer 4

7327/11 =75/11=2

Question 5

1,22,333…… FIND 150 TERM

Answer 5

use n(n+1)/2=150 hit and trail 17 answer

Question 6

a^n+b^n and if n is odd

Answer 6

then a+b is always factor of the expression

Question 7

least 3 didgit no divided by 3,5,6,7 remainder=1,3,4,5 largest no below 4000

Answer 7

lcm(3,5,6,7)=210k-2 and n is 19

Question 8

how to find no of zeros

Answer 8

in trailing factoer like 5! find 5 and 2
in genral nos=200=remove lcm

Question 9

No of factors odd

Product of factors

Answer 9

Odd factors then the number is a perfect square and middle factor is root of the number

P=n^a. Divide power by 2

Question 10

Find 1889 digit no

Answer 10

Till 2 -189

1889-189=1,700

1700/3=567. The. Number 566+100=666.
1700/3 =566.667

Then. Point is 2/3. The ans 2nd digit
Is 6

Question 11

Question n=2^5 3^45^2

Answer 11

Total factors =90. All powers +1

Even factors = don’t take 2^0 so answer =75

Odd factors=take 2^0 only the =15

Divisible by 10 = take factors 2*5
Take minimum power so remove power of zero for both number
Then and =50

Perfect square factors =take all even powers and 0=332=10

Perfect cube factors =take 3 power and 0so =4

Nit divvisble by 30=take divisible then minus =50

Question 12

When 1000 consecutive no starting form 1000!+2 are prime ?

Answer 12

Since 1001!+2 is always an even number
And for all odd number like divide by 17,19 they all there inside so divisible

Question 13

Prefect numbers

Answer 13

6,28,496 ,8128 …….

Question 14

420 factorial when divide by 13^n then maximum value of n

For another nos 24 divid 360factorial

Answer 14

Divide 420/13. Quotient =32

Remainder 32/13. Quotient =2
Then n =34 this only works in prime nos

24=2^3 * 3

No of 2=356. So we find 3 2s so 118

And 3s = 178 And =118

Question 15

Fin no of zeros in 2200! Base 38

Answer 15

38=2*19. The find pairs in 2200!

Then ans=121

Question 16

Product of factors no real no
Perfect square no

Answer 16

N= N^no of factors /2.

• no of factors of perfect square is odd
  Non perfect square is even

36=36^9/2. So 6^2*9/2. So 6^9

Question 17

Sum of reciprocal of factors

Answer 17

Sum of factors /N

Question 18

No of co primes

Answer 18

100 = prime factor 2 and 5

100*(1-1/2) *(1-1/5) =40

Sum of Cl primes less than 100
S=N*Q/2

Question 19

No of factors of 7200 perfect square

Answer 19

2^53^25^2.

No of factors = (5/2+1) (2/2+1) (2/2+1). Take quotient 322=12

Cube = divide be 3 and =2

For both divide by 6 =find ans

Question 20

Ulenus theorem

Answer 20

a^n quotient /N. r =1. A,n=co prime N natural nos

17^20|25=1

Question 21

If divide 4^n/6.

Answer 21

Then Learn r=4 always

Question 22

If 24 power odd
24 power even
76 any power

Answer 22

1- last 2 digit =24

2.last 2 digit =76

3. Last 2 digit =76

2 power 10=1024
6 power 5 =7776

Question 23

Divisible rule of 11-7-13.

Answer 23

No 2387643. Make three groups

2387643. Make three groups

002-387-643. The. 645-387=258 divide by 11-7-13 not divisible

Question 24

Abab
abcabc/

Answer 24

1-101
2.1001