How to prove this for every Arithmetic Progression?

[tellmesomething]

TL;DR

I just made some random arithmetic progressions and i noticed that the lowest common multiple of the common difference of any two arithmetic progressions is equal to the common difference of an AP generated by the common terms present in both of the APs.

If i take any two APs
say a=1; d=5;
1,6,11,16,21,26,31,36,41,46,51......1+(n-1)5.

say a=2; d=3;
2,5,8,11,14,17,20,23,26,29,32,35,38,41......2+(n-1)3.

If i pick out the common terms here, I get an AP again o common difference 15.
11,26,41....11+(n-1)15

How can i prove that the common terms of any 2 APs will give me another AP and the lowest common multiple of their common differences will be the common difference of the new AP? Any hints?

 

[Mark44]

What if there are no terms that are present in both arithmetic progressions? E.g. 1, 5, 9, 13, ... and 10, 20, 30, 40, ... For the first, $a_0 = 1$ and d = 4. For the second, $a_1 = 10$ and d = 10.

 

[tellmesomething]

What if there are no terms that are present in both arithmetic progressions? E.g. 1, 5, 9, 13, ... and 10, 20, 30, 40, ... For the first, $a_0 = 1$ and d = 4. For the second, $a_1 = 10$ and d = 10.

Oh i didn't think of this.......

 

[vela]

What if there are no terms that are present in both arithmetic progressions? E.g. 1, 5, 9, 13, ... and 10, 20, 30, 40, ... For the first, $a_0 = 1$ and d = 4. For the second, $a_1 = 10$ and d = 10.

Then the OP's claim would be vacuously true. But if there are common terms, is the claim true?

 

[PeroK]

How can i prove that the common terms of any 2 APs will give me another AP and the lowest common multiple of their common differences will be the common difference of the new AP? Any hints?

Leaving aside the trivial case where two AP's have no terms in common, let $A$ be the lowest common term. Also, we should assume that both common differences are positive. For subsequent terms, the two AP's are:
$$A, A + d_1, A + 2d_1 \dots$$and$$A, A + d_2, A + 2d_2 \dots$$Now there are two cases ... Hint $d_1, d_2$ need not be whole numbers.

 

[tellmesomething]

Leaving aside the trivial case where two AP's have no terms in common, let $A$ be the lowest common term. Also, we should assume that both common differences are positive. For subsequent terms, the two AP's are:
$$A, A + d_1, A + 2d_1 \dots$$and$$A, A + d_2, A + 2d_2 \dots$$Now there are two cases ... Hint $d_1, d_2$ need not be whole numbers.

So like say n1d1 is the common difference after which another common term is seen. And n2d2 is the common difference in the second ap after which this same common term is seen so
A+n1d1=A+n2d2
n1d1=n2d2

does this imply that n1d1 is the LCM(D1,D2)...?

 

[fresh_42]

You have essentially two equations at hand:
$$
d_1 d_2 = \operatorname{lcm}(d_1,d_2)\cdot \operatorname{gcd}(d_1,d_2)\quad\text{ and }\quad\operatorname{gcd}(d_1,d_2)=c_1d_1+c_2d_2
$$
for some integers $c_1,c_2.$

 

[PeroK]

The common difference is not necessarily an integer, or even a rational. It's possible, therefore, that the two sequences have only one term in common. But, if they have a second term in common ...

 

[fresh_42]

Another hint: if a prime factor $p$ divides $d_1$ what can we say about whether $p$ divides $d_2$, too, or not?

 

[PeroK]

Another hint: if a prime factor $p$ divides $d_1$ what can we say about whether $p$ divides $d_2$, too, or not?

Are all AP's necessarily sequences of integers?

 

[fresh_42]

Are all AP's necessarily sequences of integers?

Does it make sense to speak about $\operatorname{lcm}$ if not?

 

[PeroK]

Does it make sense to speak about $\operatorname{lcm}$ if not?

Of course not. I made the point in post #5 that $d_1, d_2$ are in general real numbers. And, in fact, the proof that the common terms form an AP is simpler in general.

 

[PeroK]

Assuming both AP's are infinite and increasing, there are three possibilities:

a) They have no terms in common.
b) They have one term in common.
c) They have an infinite number of terms in common, which form an AP

We can show a) and b) by example.

If the sequences have two terms in common, then we can choose the lowest two such terms and write them as $A$ and $A + D$. That means that $D$ is the smallest number such that $D = kd_1 + md_2$ for some positive integers $k, m$. This implies that the next term in common is $A + D + D = A + 2D$ and so on.

The common terms form an AP with first term $A$ and common difference $D$.

Note that we have case c) iff the ratio $d_1/d_2$ is rational.