Great work

Thanks. I should pay atrtention to homeworks!

Help me bro i tried many times to understand and i almost got it but i did understand in the questions no 24 in which a congruent b ( mod 5 ) before we solve it further my question is i take a = 4 , b = 7 which is not divisible by 5 and in the question there was written that a , b is integer so if i take this 2 integers in the place of a and b and it is not divisible by 5 so how would i solve it further

I dont understand 😭😭😭

I think I didn't quite understand the 1st theorem,how did you factor a+b=2n + 2k + 1 and how does it come up with 2m + 1 once again amazing lecture,I just didn't quite get it the last part.

@kanishkabhardwaj6095

2 жыл бұрын

(letting m = n + k) From the book.

@O.Mohamed.H

2 жыл бұрын

@@kanishkabhardwaj6095 and (m) is a natural number, I think?

@kanishkabhardwaj6095

2 жыл бұрын

@@O.Mohamed.H yes

@O.Mohamed.H

2 жыл бұрын

@@kanishkabhardwaj6095 Thanks❤️

So, the way that I've found the question 26 proof was that I wasn't able to factor 3 from (3k+1)^2 and (3k+2)^2 and therefore you can't write those numbers in the form n = 3r. I don't know if this is 100% right tho.

@jg394

4 жыл бұрын

Sounds like you got it.

@Rofida-rh1wq

Жыл бұрын

Could you explain it more ,?

@GuilhermeMichel

2 ай бұрын

tá certo!

I don't understand the point of exercise 26. Why include the forms (3k + 1) and (3k + 2)?

@kanishkabhardwaj6095

2 жыл бұрын

Tim Farage , Former Professor of Mathematics Answered 5 years ago · Author has 4.3K answers and 10.6M answer views When you divide any integer by 3, the remainder must be less than 3. This is really just the result of what we mean by integer division. For instance, how many cartons (each containing 12 eggs) can be made from 50 eggs? We'd divide 12 into 50 to get a quotient of 4 with a remainder of 2. So we could make 4 cartons of eggs with 2 left over for breakfast. It would not make sense to have a remainder that was 12 or more because then we'd just make more cartons until we had less than 12 left over. This can be expressed by the equation: 50 = 4 * 12 + 2,where 4 is the quotient and 2 is the remainder. So since dividing an integer n by 3 must leave a remainder less than 3, the remainder must be 0 or 1 or 2. If the remainder is 0, this can be written as the equation: n = 3k where k is the quotient, and 0 is the remainder. (You can write this as n = 3k + 0 to see the remainder explicitly). If the remainder is 1, this can be written as the equation: n = 3k + 1 where k is the quotient, and 1 is the remainder. And if the remainder is 2, this can be written as the equation: n = 3k + 2 where k is the quotient, and 2 is the remainder. This takes care of every case, thus every integer is of the form 3k, or 3k + 1, or 3k + 2. HOPE THIS HELPS

@kanishkabhardwaj6095

2 жыл бұрын

this is from quora

How do we prove exercise 2, 4, 7 ? ??

@jg394

2 жыл бұрын

Come join us on discord to discuss: discord.gg/ZBu2mDn

I don't get exercise 24 please help

@swampdigger1

2 жыл бұрын

This pdf about congruences helped me understand what was going on little better: www.math.toronto.edu/rosent/Mat246Y/PDF/cong.pdf

11:50

Iam not quite sure in 26 proof but you (3k+1) = 3r when 3k remind is 1 , when remind of 3r is 0 so are equal to k+1=r

Where's the homework?

@kanishkabhardwaj6095

2 жыл бұрын

EXERCISES 1. Give the proofs for the cases of Theorem 1 which were not proved in the text. 2. Prove: If a is even and b is any positive integer, then ab is even. 3. Prove: If a is even, then a3 is even. 4. Prove: If a is odd, then a3 is odd. 5. Prove: If n is even, then ( - l )n = 1. 6. Prove: If n is odd, then ( - l )n = -1. 7. Prove: If m, n are odd, then the product mn is odd. Find the largest power of 2 which divides the following integers. 8. 16 9. 24 10. 32 11. 20 12. 50 13. 64 14. 100 15. 36 Find the largest power of 3 which divides the following integers. 16. 30 17. 27 18. 63 19. 99 20. 60 21. 50 22. 42 23. 45 24. Let a, b be integers. Define a = b (mod 5), which we read “ a is congruent to b modulo 5” , to mean that a - b is divisible by 5. Prove: If a = b (mod 5) and x = y (mod 5), then a + x = b + y (mod 5) and ax = by (mod 5). 25. Let d be a positive integer. Let a, b be integers. Define a = b (mod d) to mean that a - b is divisible by d. Prove that if a = b (mod d) and x = y (mod d), then a + x = b + y (mod d) and ax = by (mod d). 26. Assume that every positive integer can be written in one of the forms Sk9 Sk + 1, 3& + 2 for some integer k. Show that if the square of a positive integer is divisible by 3, then so is the integer.

@kanishkabhardwaj6095

2 жыл бұрын

Homework is in the book itself

@GhanchakkarVideos

2 жыл бұрын

How do we prove exercise 2, 4, 7 ? ??