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Thank you, very well explained.

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Shouldn’t we use the least number with the most demand if it’s a tie

you have really helped me understand this concept, the explanation was very straight foward,thankyou.

A good presentation of work. Keep it up fam

how to download on Mac??

wonderfull work Sir please keep it up

Thank you bro😊😊

Thanks very much sir

very good

very clear

Very understandable 😊

Proudly Zimbabwean🎉

Just WOW

Great

Thanks

Thank you sir❤

Great video

Wów, so amazing

Good

Decide which of the following integers are divisible by 22. (a) 0. Dr M for such a question if 0 is divisible by 22 how can we conclude is since the remainder is 0 0 is divisible by 22 ok?

Good evening sir for : Show that if a and b are positive integers and a | b, then a ≤ b. how would we express this or conclude this

Prove or disprove that if a | bc, where a, b and c are positive integers and a ̸= 0, then a | b or a | c. do we use counter examples for this one

i did :(Show that if a, b, c and d are integers with a and c nonzero, such that a | b and c | d, then ac | bd.) in two ways 1) where Given a∣b, we can write: 𝑏=a⋅k(1) Given c∣d, we can write: 𝑑=c⋅m(2) then using bd ,bd=(ak)(cm)=a⋅c⋅k⋅m. then we let x=km then we have bd=ac⋅x proving it but im not entirely sure about this approach, The second approach i took was a longer one where i said for ac | bd, bd=ac⋅z then from (1) and (2) i took the b and d equivalent (ak)⋅(cm)=(b/k)⋅(d/m)⋅z , this approach by substituting actually eliminated k and m leaving bd=ac⋅z

How about :Show that if a, b, c and d are integers with a and c nonzero, such that a | b and c | d, then ac | bd , I tried solving for the third by substituting the 1st and 2nd equations into it but the way its turning out im not sure how it would be showing that the statement is true

Sorry to bother this late, how about :Show that if a, b, c and d are integers with a and c nonzero, such that a | b and c | d, then ac | bd

Good evening Dr M, Question 1 on the worksheet says :Does 18 divide each of these numbers? with emphasis to the word 'divide' would we use if a|b ∃c∈Z : b=ac as a solution. The reason i ask this , lets say for 1.c) 1005. of which 18 does not divide 1005 that i know so here divide would be the same as 18|b?

Very good. I appreciate your efforts

Dr M,do we have to rewrite the whole matrix or can we do just as you did by placing the brackets in the table

Can you explain how you expanded it?

Nice

As an aside, if you are only looking for primitive roots, then it's often quicker to check the _largest_ factors of ϕ(m) first. Taking the example of m=13, we have ϕ(13) = 12. We know that r^12 ≡ 1 mod 13 (by Fermat's Little Theorem), so we don't need to check that. That means that r^6 must be congruent to 1 or -1 mod 13 (since r^6 is the square root of r^12). If r^6 ≡ 1, then r is not a primitive root. If r^6 ≡ -1 ≡ 12 mod 13, then we need to check r^4, but we don't need to check any further since none of r^1, r^2, r^3 can possibly be congruent to 1 if r^4 and r^6 are not.

Thanks for the explanation

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Is this maximization?

Very helpful 🙏

i wanted to ask if we are going to sketch the graphs on the test on Friday

Even 6 divides a³-a

For checking a transitive relation in the form x>y do we use number examples such as above and if we do how would we use the number examples Can we use any combinations that can give us 15 and if lets say the numbers we choose are (8,7) for a>b its true but for b>c (7,8) is not true how would we tackle such a question saying x>y defines a relation on N determine if its reflexive, symmetric and transitive. On worksheet 2 .2b the question asks xy is a square of an integer determine the relations, how do we work it out

What is the Criteria for symmetric nature in the first set, then transitive nature in the second set how is it obtained because (3,3) and (4,4) don't satisfy

I am in grade 10 and I only undrstood what is Integer,Natural no and odd no.😢