# Short method for computing this?

• mohamed el teir
In summary, to find the least positive integer value for q that makes the side of equation 26q+1 divisible by 7, you can work in mod 7. For example, if 11x+8 is divisible by 5, then in mod 5, 11x+8 = 0 and when working with a higher mod like mod 26, you can use the same method, but it may not always be efficient. Other methods, such as searching for 0 in a column, may also be used.
mohamed el teir
suppose a side of an equation : 26q+1 , and i want the least positive integer value for q that makes this side of equation divisible by 7, is there any short method to do this ?

I don't know any short method, but you can work in mod 7.

So for eg. if 11x + 8 is divisible by 5, then in mod 5
11x + 8 = 0
10x + x + 8 = 0
0 + x + 8 = 0
x + 3 + 5 = 0
x + 3 + 0 = 0
So x is 2, 7, 12, etc.

Ed: just noticed this is too simple. So 13x +9 is divisible by 5. In mod 5,
13x + 9 = 0
3x + 4 = 0
3x = -4
3x = 1
so 3x = 6, 11, 16, 21, 26, 31, 36, etc
so x= 2, 7, 12, etc.

Last edited:
Merlin3189 said:
I don't know any short method, but you can work in mod 7.

So for eg. if 11x + 8 is divisible by 5, then in mod 5
11x + 8 = 0
10x + x + 8 = 0
0 + x + 8 = 0
x + 3 + 5 = 0
x + 3 + 0 = 0
So x is 2, 7, 12, etc.

Ed: just noticed this is too simple. So 13x +9 is divisible by 5. In mod 5,
13x + 9 = 0
3x + 4 = 0
3x = -4
3x = 1
so 3x = 6, 11, 16, 21, 26, 31, 36, etc
so x= 2, 7, 12, etc.
but when working with higher mod like mod 26 would this way be efficient ? for example when 9x-1 is divisible by 26

No, I don't know how to make that simpler. Perhaps if you search on Diophantine equations, you can find something to help.

AFAIK
9x-1 = 0 (Mod 26)
then 9x= 1 is as far as I can get.
So 9x = 1, 27, 53, 79, 105,... The only way I can see to help now, is to look at these Mod 9
(so 0 = 1, 0, 8, 7, 6, 5, 4, 3, 2, 1, 0, ..) so the second term is the first that is not false
so 9x = second term = first term + 1 x 26 = 1 + 1*26 = 27
so x= 3
Check 9 x 3 - 1 = 27 -1 =26

That was a bit too easy,
so try 9x -17 = 0 (Mod 26)
then 9x = 17
So 9x = 17, 43, 69, ...
now to looking at these Mod 9
(so 0 = 8, 7, 6, 5, 4, 3, 2, 1, 0, .. ) so the 9th term is the first that is not false
so 9x = 9th term = first term + 8 x 26 = 17 + 8*26 = 225
so x= 25
(Check 9 x 25 - 17 = 225 -17 = 208 = 26 x 8 )

It actually looks as if you could simplify the Mod 9 step, but in some cases the change in mod base does not give such a simple sequence.

A short trip into Excel gave q=4 (next one is 11, then 18 , 25).

Svein said:
A short trip into Excel gave q=4 (next one is 11, then 18 , 25).
But what method did you use in Excel?

Brute force - I created the series 0, 1, 2... 29 in column B, 26*Bn + 1 in column C and Mod(Cn, 7) in column D. Then I just looked for 0 in column D.
(
0 1 1
1 27 6
2 53 4
3 79 2
4 105 0
5 131 5
6 157 3
7 183 1
8 209 6
9 235 4
10 261 2
11 287 0
12 313 5
13 339 3
14 365 1
15 391 6
16 417 4
17 443 2
18 469 0
19 495 5
20 521 3
21 547 1
22 573 6
23 599 4
24 625 2
25 651 0
26 677 5
27 703 3
28 729 1
29 755 6)

## 1. What is the short method for computing this?

The short method for computing involves using a simplified approach to solve a problem or equation. It is often a faster and more efficient method compared to traditional methods.

## 2. When should I use the short method for computing?

The short method for computing can be used in various situations, but it is most commonly used for complex calculations or when time is limited. It is also useful for checking answers obtained through other methods.

## 3. How do I learn the short method for computing?

Learning the short method for computing involves understanding the basic principles and concepts behind the method and practicing it through various problems and equations. There are also many online resources and tutorials available for learning the method.

## 4. Does the short method for computing always give accurate results?

Yes, the short method for computing follows the same principles and formulas as traditional methods, so it will always give accurate results as long as the steps are followed correctly.

## 5. Are there any disadvantages to using the short method for computing?

One possible disadvantage of the short method for computing is that it may not work for every type of problem or equation. It is important to understand when it is appropriate to use the short method and when other methods may be more suitable.

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