Explanation : 

Problem: Find the digit P in 3798125P369 so that the number is divisible by 7

Step 1: Divide into Three-Digit Blocks (from right to left)
– Write as: 037, 981, 25P, 369
– This splits the number into blocks of three: 037|981|25P|369

Step 2: Apply Alternating Addition and Subtraction Rule
– Start from rightmost block: 369
– Then alternate: 369 – 25P + 981 – 037
– Note: This is based on the divisibility rule of 7 where we alternate adding and subtracting blocks of 3 digits

Step 3: Simplify the Expression
369 – (250 + P) + 981 – 37
= 369 – 250 – P + 981 – 37
= 1063 – P

Step 4: Solve for P Using Modulo 7
– For the number to be divisible by 7:
– 1063 – P must be divisible by 7
– 1063 ÷ 7 = 151 remainder 6
– So, 1063 ≡ 6 (mod 7)
– Therefore: 6 – P ≡ 0 (mod 7)
– This means P = 6

Answer: P = 6

Explanation : 

Problem: Find the remainder when 3^2019 is divided by 10

Let’s solve this step by step:

1) To find 3^2019 mod 10, examine the pattern of remainders when successive powers of 3 are divided by 10:
3^1 = 3
3^2 = 9
3^3 = 27 ≡ 7 (mod 10)
3^4 = 81 ≡ 1 (mod 10)
3^5 = 243 ≡ 3 (mod 10)
3^6 = 729 ≡ 9 (mod 10)
3^7 ≡ 7 (mod 10)
3^8 ≡ 1 (mod 10)

2) The pattern of remainders is: 3, 9, 7, 1, 3, 9, 7, 1, …
This forms a cycle of length 4

3) For 3^2019:
2019 = 504 × 4 + 3
Thus 3^2019 will have the same remainder as 3^3

4) Since 3^3 ≡ 7 (mod 10)

Therefore, when 3^2019 is divided by 10, the remainder is 7.

The answer is (c) 7.

Explanation : 

Integers are listed from 700 to 1000

The sum of the digits of an integer is 10 if the digits add up to 10. For example, the number 703 has a digit sum of 7 + 0 + 3 = 10. We can find all integers between 700 and 1000 that have a digit sum of 10 by listing them out:

703, 712, 721, 730, 802, 811, 820, 901, 910.

There are 9 integers between 700 and 1000 that have a digit sum of 10. So the correct answer is (d) 9.

Explanation : 

A boy plays with a ball

Starting height = 1.5 meters
Bounce ratio = 4/5 (ball reaches 4/5th of previous height)
Stop condition = height < 0.5 meters (50 cm)

Height Calculations:

1. Initial Drop: 1.5 meters

2. First Bounce:
Height = 1.5 × 4/5 = 1.2 meters

3. Second Bounce:
Height = 1.2 × 4/5 = 0.96 meters

4. Third Bounce:
Height = 0.96 × 4/5 = 0.768 meters

5. Fourth Bounce:
Height = 0.768 × 4/5 = 0.6144 meters

6. Fifth Bounce:
Height = 0.6144 × 4/5 = 0.49152 meters
(This is less than 50 cm, so ball stops)

Result:
– Ball hits ground 5 times before stopping
– Final height (0.49152 m) is below stopping condition (0.5 m)
– Answer: 5 bounces

Explanation : 

A student appeared in 6 papers

– Total marks per subject = 100
– Total subjects = 6
– Total possible marks = 600
– Overall score = 60% = 360 marks
– Marks distribution pattern: 5x, 6x, 7x, 8x, 9x, 10x

Step 1: Form equation
5x + 6x + 7x + 8x + 9x + 10x = 360
45x = 360

Step 2: Solve for x
x = 360/45 = 8

Step 3: Calculate marks in each subject
Subject 1: 5x = 5 × 8 = 40 marks
Subject 2: 6x = 6 × 8 = 48 marks
Subject 3: 7x = 7 × 8 = 56 marks
Subject 4: 8x = 8 × 8 = 64 marks
Subject 5: 9x = 9 × 8 = 72 marks
Subject 6: 10x = 10 × 8 = 80 marks

Step 4: Count subjects below 60%
60% of 100 = 60 marks
Subjects below 60 marks:
1. 40 marks
2. 48 marks
3. 56 marks

Therefore: Student scored less than 60% in 3 subjects.

Explanation :

There are two classes a and b

Class A (Initial):
– Total students: 25
– Highest score: 21
– Lowest score: 17
– Four students will be shifted out

Class B (Initial):
– Total students: 30
– Highest score: 30
– Lowest score: 22
– Will receive four students from Class A

Analysis:

1. Effect on Class B’s Average:
– Initial students: 30
– New total: 34 students
– All incoming students have scores ≤ 21
– All original students have scores ≥ 22
– Therefore, adding any four students from Class A will definitely lower Class B’s average

2. Effect on Class A’s Average:
– Initial students: 25
– New total: 21 students
– Cannot determine if average will increase because:

• If highest scoring students (around 21) are shifted out → average will decrease
• If lowest scoring students (around 17) are shifted out → average will increase
• Without knowing which students are shifted, change in average is uncertain

Conclusion:
– Class B’s average will definitely decrease
– Class A’s average could increase or decrease depending on which students are shifted

Explanation:

Sum of 3-Digit Numbers with Digits 3, 6, and 9

Given Digits: 3, 6, 9

Step 1: List all possible 3-digit combinations (without repetition)
1. 369
2. 396
3. 639
4. 693
5. 936
6. 963

Step 2: Calculate sum (S)
369 + 396 + 639 + 693 + 936 + 963 = 3996

Step 3: Verify divisibility by 74
3996 ÷ 74 = 54
Therefore, 3996 is divisible by 74

Step 4: Verify divisibility by 9
3996 ÷ 9 = 444
Therefore, 3996 is divisible by 9