Divisibility Rule for 3 & 9

Q.

An eight digit number divisible by 9 is to be formed by using 8 digits out of the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 without replacement.
The number of ways in which can be done is

1. 4(7!)

2. 2(7!)

3. (36)(7!).

4. (7!)(33)

Q. Digital sum of a number is obtained by adding all the digits of a number until a single digit is obtained. Find the digital sum of ${19}^{100}$.

1. 7
2. 1
3. 4
4. 9

Q. What is the least number which, when divided by 5, 6, 7 and 8 gives the remainder 3 but is divisible by 9?

1. 1683
2. 1793
3. 1573
4. None of these
5. 1463

Q. How many numbers between 1 and 200 are exactly divisible by exactly two of 3, 9 and 27?

1. 14
2. 16
3. 17
4. 15

Q.

Let the least number of six digits, which when divided by 4, 6, 10 and 15, leaves in each case the same remainder of 2, be N. The sum of the digits in N is :

1. 6

2. 3

3. 4

4. 7

5. 5

Q. $\frac{{32}^{{32}^{32}}}{9}$ will leave a remainder

1. 4
2. 7
3. 1
4. 2

Q. The remainder when $2^2+{22}^2+{222}^2+{2222}^2+....(222....49twos{)}^2$ is divided by 9 is

1. 5
2. 2
3. 6
4. 7

Q. How many numbers lying between 1000 and 10000 can be formed by using the digits 1, 3, 5, 6, 7, 8, 9, no digits being repeated?

1. 640
2. 940
3. 840
4. 740
5. None of these

Q. Let S be a set of positive integers such that every element n of S satisfies the conditions:
I.$1000\le n\le 1200$
II. Every digit in n is odd
Then how many elements of S are divisible by 3?

1. 9
2. 10
3. 11
4. 12

Q.

How many numbers of $6$ digits can be formed from the digits of the number $112233$?

1. $30$

2. $60$

3. $90$

4. $120$

Q.

The remainder when $1!+2!+3!+4!+......+100!$ is divided by $240$ is?

1. $153$

2. $33$

3. $73$

4. $187$

Q. If $1826X73$ is divisible by 9, then what can be the value of 'X'?

1. 9
2. 0
3. 3
4. (a) or (b)

Q.

What will be the least number which when doubled will exactly be divisible by 12, 18, 21 and 30?

1. None of these

2. 196

3. 630

4. 1260

5. 2520

Q. How many 5 digit numbers are divisible by 3 and contain the digit 6?

1. 8776

2. 12503

3. none of these

4. 7499

Q. What will be the value of x for $\frac{({100}^{17}-1)+({10}^{34}+x)}{9}$; the remainder =0?

1. 3
2. 6
3. 9
4. 8

Q. Four distinct numbers are selected from first 100 natural numbers such that each number is divisible by both 3 and 5. The number of selections possible is
___

Q. abcde is a five digit number when multiplied by 13 it gives a number, purely formed by the digit 9. Then the value of a + b + c + d + e is:

1. divisible by 8
2. equal to 27
3. divisible by 11
4. all of these

Q.

The least multiple of 7, which leaves a remainder of 4, when divided by 6, 9, 15 and 18 is:

1. 74

2. 94

3. 184

4. 364

5. None

Q. The remainder when $(3{)}^{67!}$ is divided by 80:

1. can't be determined
2. 0
3. 1
4. 2

Q.

Choose the most appropriate option to replace (?).
34, 18, 10, ?

1. 8
2. 7
3. 6
4. None of the above
5. 5

Q. In a nuclear power plant, a technician is allowed for an interval of maximum 100 minutes. A timer with a bell rings at specific intervals of time such that the minutes when the timer rings are not divisible by 2, 3, 5 and 7. The last alarm rings with a buzzer to give time for decontamination of the technician. How many times will the bell ring within these 100 minutes and what is the value of the last minute when the bell rings for the last time in a 100 minute shift?

1. 25 times, 89
2. 21 times, 97
3. 22 times, 97
4. 19 times, 97

Q.

A prime number x greater than 100 leaves a remainder y on division by 26. How many values can y take?

1. 8

2. 9

3. 7

4. 6

Q.

Write the odd numbers between $36and53$

Q. Digital sum of a number is obtained by adding all the digits of a number until a single digit is obtained. What is the digital sum of ${4444}^{4444}$?

1. 3
2. 7
3. 2
4. 6

Q. The number from 1 to 33 are written side by side as follows: 123, 456 $\dots$ 33. What is the remainder when this number is divided by 9?

1. 0
2. 6
3. 1
4. 3

Q. Find the sum of the sum of the sum of digits (i.e. digit sum) of 25! (e.g., ) digit sum of 378 = 3+7+8 = 18 : 1+8 = 9

1. 7
2. 5
3. 6
4. 9
5. 3

Q. Four distinct numbers are selected from first 100 natural numbers. How many numbers are there which are divisible by both 3 and 5?

1. 4
2. 6
3. 7
4. 2

Q. Suppose the seed of any positive integer n is defined as follows: Seed $\left(n\right)=n,$ if $n<10=$seed$\left(s\right(n\left)\right)$, otherwise, Where $s\left(n\right)$ indicated the sum of digits of n. For example, seed(7)=7, seed(248)=2+4+8=seed(14)=seed(1+4)=seed(5)=5, etc. How many positive integers n, such that n

1. 39
2. 72
3. 81
4. 108
5. 55

Q. 2 9 32 105 436 2159 13182

1. 436
2. 2195
3. 32
4. 9
5. None of these

Q. The remainder when $(2^{54}-1)$ is divided by 9 is:

1. 0
2. 8
3. 7
4. none of these