Question

When 'n' is divided by 'd' it leaves a remainder 9, when square of the number 'n' is divided by the same
divisor'd' it leaves a remainder 1. Find the number of possible values for 'd'.
Select one:
a. 4
O b. 8
O c. 10
O d. 5​

Answer 1

Step-by-step explanation:

n=(k∗d)+9 ->(given)

Squaring on Both Sides , we get,

n²=(k∗d+9)²

=k²d²+2∗(k∗d)∗(9)+9²

=d(k²d+18k)+81

Assume y=k²d+18k

n²=d∗(y)+81-> 1

n²=d∗(x)+1-> 2 (given)

From 1 & 2, we get

d∗(y)+81=d∗(x)+1

d∗(x−y)=80

d∗z=80(x−y=z)

So, Number of values d can take is nothing but factors of 80 which include 1,2,4,5,8,10,16,20,40,80, a total of 10 values

or simply 80=(24)∗(51)

Number of Factors=(4+1)*(1+1)=5*2=10.

But since d leaves a remainder of 9 , So, d>9

So, possible values of d are 10,16,20,40,80.