Got it 👍 You want a comprehensive 3000-word write-up on Remainders. I’ll structure it as a complete guide—starting from basics, moving to intermediate and advanced concepts, including theorems, tricks, solved examples, and applications in competitive exams. Here’s the structure I’ll follow: 📑 Outline of the 3000-Word Write-up on Remainders 1. Introduction to Remainders 2. Basic Concepts and Definitions Dividend, Divisor, Quotient, and Remainder Properties of Division 3. Rules and Properties of Remainders Addition, Subtraction, Multiplication with Remainders Division and Powers with Remainders 4. Special Theorems Remainder Theorem (Algebra) Fermat’s Little Theorem Euler’s Theorem Wilson’s Theorem Chinese Remainder Theorem 5. Shortcut Techniques for Competitive Exams Finding last digits Remainders in factorials Remainders with large powers 6. Step-by-Step Solved Examples Easy, Moderate, and Advanced Levels 7. Applications of Remainders Aptitude Exams (CAT, GATE, TCS NQT, etc.) Computer Science (modulo operations, hashing) Cryptography 8. Common Mistakes and Misconceptions 9. Practice Questions with Solutions 10. Conclusion and Tips --- Would you like me to write the entire 3000 words in one single response (very long) or break it into parts (for easier reading and structuring, e.g., 5–6 parts)?

1. Navigating Directions and Mastering
Distances
Lecture Explanation
Presented by
Mr. Rajendra Bhanu Teja– Aptitude Trainer

2. CONCEPT

3. Surya starts walking towards North, turns left , again
turns left , again turns right , again turns left , again
turns right , again turns right . In which direction is he
walking now ?
a. North
b. South
c. West
d. South- East
Model-1: Facing/ Moving Direction
Question Solution
Presented by
Mr. Rajendra Bhanu Teja– Aptitude Trainer

4. Question Solution
Facing to the South Mr. Karthik turns in certain ways,
which of the following turns will not lead him to the
same side ?
a. Left , Left , Right ,Left , Left , Right
b. Left , Right , Left , Right , Left , Right
c. Right , Right , Left , Right , Left , Left
d. Left , Right , Right , Left , Right , Left
Model-1: Facing/ Moving Direction
Presented by
Mr. Rajendra Bhanu Teja– Aptitude Trainer

5. Model-2: Staring Direction
Question Solution
Raju after walking 25 m turns left , turns left , turns
right , turns left , turns right , turns left , finally turns
left . In the end , he was moving towards North . From
which direction was he start his journey ?
a. West
b. North
c. South
d. East
Presented by
Mr. Rajendra Bhanu Teja– Aptitude Trainer

6. Model-2: Staring Direction
Question Solution
After walking for 1 m, Madhuri turned to her left and
walked another 12 m. Again, she turned to her left and
walked for 4 m. Now she is travelling in East direction.
In which direction did she start her journey?
a. West
b. East
c. North
d. South
Presented by
Mr. Rajendra Bhanu Teja– Aptitude Trainer

7. Question Solution
Model-3: Distance w.r.t Direction
Prasanth walked 40 metres towards South and moves
70 m towards East and then moves a distance of 40 m
towards South and then moves 70 m towards West . In
which direction and how long distance he is now from
the starting point ?
a. 60 East
b. 80 South
c. 70 South
d. 120 East
Presented by
Mr. Rajendra Bhanu Teja– Aptitude Trainer

8. Question Solution
Model-3: Distance w.r.t Direction
I drove 50 km towards east from a city ‘S’ and then
turned right and drove another 30 km. Now I turned to
my left & drove another 30 km. Finally, I turned my
right & drove 30 km to reach a city ‘F’. Find the
shortest straight distance between cities S and F.
A) 20 km
B) 25 km
C) 40 km
D) 100 km
Presented by
Mr. Rajendra Bhanu Teja– Aptitude Trainer

9. Question
Rajesh walked 20 m towards north. Then he turned right
and walks 30 m. Then he turns right and walks 35 m.
Then he turns left and walks 15 m. Finally, he turns left
and walks 15 m. In which direction and how many metres
is he from the starting position?
A) 15 m West
B) 30 m East
C) 30 m West
D) 45 m East
Solution
Model-4: Finding of Distance & Direction
Presented by
Mr. Rajendra Bhanu Teja– Aptitude Trainer

10. Question Solution
Model-4: Finding of Distance & Direction
A cyclist goes 30 km to North and then turning to East
he goes 40 km. Again, he turns to his right and goes 20
km . After this, he turns to his right and goes 40 km. in
which direction and how long distance is starting point
from the ending point ?
a. 0 km, South
b. 10 km, North
c. 10 km, South
d. 20 km, North
Presented by
Mr. Rajendra Bhanu Teja– Aptitude Trainer

11. Question Solution
A, B, C, D and E are five friends standing in a park. The
distance between A and C and that of B and C are same.
E and C are 45m apart in west-east direction. A is
towards west of C at a distance of 25m from E. C is in
between B and D and in north direction with B.
9. If the distance between B and D is 35m, then
calculate the distance between A and D.
A) 15 B) 20 C) 25 D) 30
Model-4: Finding of Distance & Direction
Presented by
Mr. Rajendra Bhanu Teja– Aptitude Trainer

12. Question Solution
Model-5: Angle
A man is facing north-west. He turns 900
in the
clockwise direction, then 1800
in the anticlockwise
direction and then another 900
in the same direction.
Which direction is he facing now?
A) South
B) South-west
C) South-east
D) East
Presented by
Mr. Rajendra Bhanu Teja– Aptitude Trainer

13. Question Solution
Model-5: Angle
Ravi is facing south-west. He turns 135° to his
clockwise and again turns 225° in anticlockwise
direction. From there, he turns 90° in clockwise
direction. Finally, he turns 45° to his anticlockwise.
Which direction is he facing now?
A) South
B) South-East
C) West
D) North-West
Presented by
Mr. Rajendra Bhanu Teja– Aptitude Trainer

14. Question Solution
Model-6: Direction Change
South-East becomes North and North-East becomes
West and all the rest directions are changed in the
same manner, then what will be the direction for the
West ?
a. North-East
b. South
c. South-East
d. South-West
Presented by
Mr. Rajendra Bhanu Teja– Aptitude Trainer

15. Question Solution
Model-7: Clock Model
Rahul put his timepiece on the table in such a way
that at 6 P.M. hour hand points to North. In which
direction the minute hand will point at 9.15 P.M.?
A) South-East
B) South
C) North
D) West
Presented by
Mr. Rajendra Bhanu Teja– Aptitude Trainer

16. Question Solution
Presented by
Mr. Rajendra Bhanu Teja– Aptitude Trainer
Model-8: Shadow Based
One morning after sunrise, Suresh was standing
facing a pole. The shadow of the pole fell exactly to
his right. To which direction was he facing?
A) East
B) West
C) South
D) Data is inadequate
Presented by
Mr. Rajendra Bhanu Teja– Aptitude Trainer

17. Question Solution
Presented by
Mr. Rajendra Bhanu Teja– Aptitude Trainer
Model-8: Shadow Based
One evening just before sunset two friends Sanju and
Manju were talking to each other face to face. If
Manju’s shadow was exactly to her left side, which
direction was Sanju facing?
A) North
B) South
C) West
D) Data inadequate

18. Extra Questions

19. Question Solution
Presented by
Mr. Rajendra Bhanu Teja– Aptitude Trainer
North-West becomes East and South becomes North-
West and all the rest directions are changed in the
same manner, then what will be the direction for the
West ?
a. North-East
b. South
c. South-East
d. South-West

20. Question Solution
Presented by
Mr. Rajendra Bhanu Teja– Aptitude Trainer
A clock showing 2:00 pm. If the hour hand is in
North-East direction. Then , what will be the direction
of hour hand, when the time of clock becomes 9:30
pm ?
a. South
b. South-East
c. North-West
d. South-West

21. Question Solution
Presented by
Mr. Rajendra Bhanu Teja– Aptitude Trainer
Arvind starts from a point A and travels 3 km
Eastwards to reach B and then turns left ,travels thrice
that distance to reach C. He again turns left and
travels five times the distance he covered between A
and B and reached his destination D. The shortest
distance between the starting point and the destination
is:
a. 17 km
b. 18km
c. 15 km
d. 12 km

22. Thank You