Given:
A number when divided by 780 gives remainder 38.
To find:
The reminder that would be obtained by dividing same number by 26.
Solution:
According to Euclis' division algorithm,
[tex]a=bq+r[/tex] ...(i)
Where, q is quotient and [tex]0\leq r<1[/tex] is the remainder.
It is given that a number when divided by 780 gives remainder 38.
Substituting [tex]b=780,\ r=38[/tex] in (i), we get
[tex]a=(780)q+38[/tex]
So, given number is in the form of [tex]780q+38[/tex], where q is an integer.
On dividing [tex]780q+38[/tex] by 26, we get
[tex]\dfrac{780q+38}{26}=\dfrac{780q}{26}+\dfrac{38}{26}[/tex]
[tex]\dfrac{780q+38}{26}=30q+\dfrac{26+12}{26}[/tex]
[tex]\dfrac{780q+38}{26}=30q+\dfrac{26}{26}+\dfrac{12}{26}[/tex]
[tex]\dfrac{780q+38}{26}=30q+1+\dfrac{12}{26}[/tex]
Since q is an integer, therefore (30q+1) is also an integer but [tex]\dfrac{12}{26}[/tex] is not an integer. Here 26 is divisor and 12 is remainder.
Therefore, the required remainder is 12.
