MHB If c|ab and gcd(b, c) = 1 why does c|ac?

  • Thread starter Thread starter MI5
  • Start date Start date
AI Thread Summary
If c divides ab and the greatest common divisor of b and c is 1, then it follows that c divides a. The proof relies on the relationship between the products ab and ac, demonstrating that since c divides ab, it must also divide ac. The discussion highlights some confusion regarding the definition of the divisibility relation. Clarification is sought on the concept of divisibility, indicating a need for a deeper understanding of the theorem. Overall, the theorem is affirmed, emphasizing the importance of grasping the foundational definitions in number theory.
MI5
Messages
8
Reaction score
0
Theorem: If c|ab and (b, c) = 1 then c|a.

Proof: Consider (ab, ac) = a(b, c) = a. We have c|ab and clearly c|ac so c|a. It's not so clear to me why c|ac. Perhaps I'm missing something really obvious.
 
Mathematics news on Phys.org
MI5 said:
It's not so clear to me why c|ac.
This is because ac = a * c, i.e., by the definition of the | (divides) relation.
 
Evgeny.Makarov said:
This is because ac = a * c, i.e., by the definition of the | (divides) relation.
I still don't understand I'm afraid. Could you say bit more please?
 
MI5 said:
I still don't understand I'm afraid. Could you say bit more please?
I need to be sure you know the definition of the relation denoted by |. Could you write this definition?
 
WOW! All I can say is thanks. (Giggle)
 
Thread 'Video on imaginary numbers and some queries'

Thread 'Video on imaginary numbers and some queries'

Hi, I was watching the following video. I found some points confusing. Could you please help me to understand the gaps? Thanks, in advance! Question 1: Around 4:22, the video says the following. So for those mathematicians, negative numbers didn't exist. You could subtract, that is find the difference between two positive quantities, but you couldn't have a negative answer or negative coefficients. Mathematicians were so averse to negative numbers that there was no single quadratic...
View full post »

Thread 'What Exactly is Dirac’s Delta Function? - Insight'

Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
View full post »

Thread 'Non-orthogonal bases'

Suppose ,instead of the usual x,y coordinate system with an I basis vector along the x -axis and a corresponding j basis vector along the y-axis we instead have a different pair of basis vectors ,call them e and f along their respective axes. I have seen that this is an important subject in maths My question is what physical applications does such a model apply to? I am asking here because I have devoted quite a lot of time in the past to understanding convectors and the dual...
View full post »

Similar threads

MHB Does Commutativity Hold for Matrices A and B with a Specific Matrix C?
Replies
3
Views
2K
I Are there any two pairs of integers with the same result in a specific function?
Replies
7
Views
2K
MHB Simplify a+b+c+d+2(ab+ac+ad+bc+bd+cd)+4(abc+abd+acd+bcd)+8(abcd)
Replies
3
Views
2K
MHB Find the sum of a^(1/3)+b^(1/3)+c^(1/3)
Replies
2
Views
2K
MHB Geocaching: Solve a+b=c c/b=b/a Problem
Replies
4
Views
1K
MHB Can AC^2 be Proven to Equal AB(AB+BC) in Triangle ABC with Angles B=80 and C=40?
Replies
3
Views
2K
MHB Solving for $a\times b \times c$: Equations (1), (2), and (3)
Replies
6
Views
2K
B Missing out on intermediate algebraic steps
Replies
23
Views
2K
MHB Minimum of $a$ for $a^2+2b^2+c^2+ab-bc-ac=1$
Replies
1
Views
1K
I On a proof of the converse of the supporting hyperplane theorem
Replies
9
Views
2K

Hot Threads

Recent Insights

Back
Top