3. How many such 4 consecutive numbers are there less than 1000 when added gives a sum which is divisible by 10? For example, 16 +17+18+19=70.

Answer:

Explanation:

Let the numbers be n, n+1, n+2, n+3.

Sum of these numbers = 4n + 6.

Let 4n + 6 = 10k

So n = 10k−6410k−64 = 2k−1+2k−242k−1+2k−24 = 2k−1+k−122k−1+k−12

So k takes only odd numbers.

For k = 1, n = 1

For k = 3, n = 6

For k = 5, n = 11

. . . .

. . . .

Final value of n = 996.

So total values = =l−ad+1=996−15+1=l−ad+1=996−15+1 = 200

4. 1,2,3,4,5,6,7 are arranged such that sum of two successive numbers is a prime number. For example, 1234765 (i.e. 1+2=3, 2+3=5, 3+4=7….)

1. How many such possible combinations occur?

2. How many possible combination occurs if first number is 1/7 and last number is 7/1 (i.e 1xxxxx7 or 7xxxxx1)?

3. How many numbers will come on 4th position(xxx_xxx)?

Explanation:

We have to follow a systematic approach for this question. We know that if at all two consecutive numbers be prime, they should not be even in the first place. So we arrange even numbers in even places, odd numbers in odd places.

2 should not have 7 as its neighbor

4 should not have 5 as its neighbor

6 should not have 3 as its neighbor

Case 1:

__ 2 ____ 4 ____ 6 __

x7 x7,5 x5,3 x3

So fix, 3 or 1 in between 2 and 4. We get the following options with reverse case also.

5234167, 7614325

5234761, 1674325

1234765, 5674321

3214765, 5674123

Case 2:

__ 2 ____ 6 ____ 4 __

x7 x7,3 x3,5 x5

5216743, 3476125

3256147, 7416523

3256741, 1476523

1256743, 3476521

Case 3:

__ 4 ____ 2 ____ 6 __

x5 x5,7 x7,3 x3

3412567, 7652143

7432165, 5612347

7432561, 1652347

1432567, 7652341

Answer 1: Total possibilities are 24.

Answer 2: 4 possibilities.

Answer 3: 3 possibilities.

5. Find the remainder when 4*4! + 5*5! + 6*6! +……………….+ 19*19! is divided by 64.

1) 32

2) 38

3) 40

4) Cannot be determined.

Explanation:

8*8! and its subsequent terms are exactly divisible by 64. As highest power of 2 in 64 is 6 and highest power of 2 in 8! is 7 which is greater than 6. So we have to find the remainder when 4*4! + 5*5! + 6*6! + 7*7! is divided by 64 which is 40.