Question

Is there an integer that has a remainder of 2 when it is divided by 5 and a remainder of 3 when it is divided by 6?

a. Is there an integer n such that n has_______?
b. Does there exist such that if n is divided by 5 the remainder is 2 and if _______?​

Answer 1

27 is an integer that has a remainder of 2 when it is divided by 5 and a remainder of 3 when it is divided by 6.

Thank You For Asking.

Hopefully u appreciate my work and

Mark this as Brainliest & get some extra Brainly Points.

Kindly Follow.

Answer 2

Answer:

27,57,87,117,147,177,207…

Step-by-step explanation:

Let n be a number which satisfies these conditions.

We have:

n≡2 (mod 5)  and  n≡3 (mod 6)  

In other words,  n=5p+2=6q+3  

When we divide  6q+3  by  5 , we have a remainder of  q+3 , since  6q+3=5q+(q+3)  and  5q  is divisible by  5 .

Therefore,  q+3≡2 (mod 5) , and so  q≡−1≡4 (mod 5) .

With this, we can write q as a number of the form  5r+4 .

Plugging this into  n=6q+3 , we have  n=6∗(5r+4)+3=30r+27  

If we divide  30r+27  by  5 , we get a remainder of  2 : note that  30r+27=30r+25+2=5(6r+5)+2 .

And if we divide it by  6 , we get a remainder of  3 ; by the same token,  30r+27=30r+24+3=6(5r+4)+3 .

So, our number list goes:  27,57,87,117,147,177,207…