This is a Test of Mathematics Solution Subjective 43 (from ISI Entrance). The book, Test of Mathematics at 10+2 Level is Published by East West Press. This problem book is indispensable for the preparation of I.S.I. B.Stat and B.Math Entrance.
Also visit: I.S.I. & C.M.I. Entrance Course of Cheenta
Show that the equation has no integral root for any integer n.
We note that implies
is divisible by 7. This implies x is divisible by 7 (as 7 is a prime number). Suppose x= 7x'. Hence we can rewrite the given equation as:
.
Cancelling out a 7 we have . Since 7 divides left hand side, it must also divide the right hand side. Since 7 cannot divide 2, it must divide
as 7 and 2 are coprime. Note that 7 cannot divide
as square of a number always gives remainder 0, 1, 4, 2 when divided by 7 and never 6. But if
is divisible by 7 then
must give remainder 6 when divided by 7. Hence contradiction.
Necessary Lemma: square of a number always gives remainder 0, 1, 4, 2 when divided by 7
Key Ideas: Modular Arithmetic
This is a Test of Mathematics Solution Subjective 43 (from ISI Entrance). The book, Test of Mathematics at 10+2 Level is Published by East West Press. This problem book is indispensable for the preparation of I.S.I. B.Stat and B.Math Entrance.
Also visit: I.S.I. & C.M.I. Entrance Course of Cheenta
Show that the equation has no integral root for any integer n.
We note that implies
is divisible by 7. This implies x is divisible by 7 (as 7 is a prime number). Suppose x= 7x'. Hence we can rewrite the given equation as:
.
Cancelling out a 7 we have . Since 7 divides left hand side, it must also divide the right hand side. Since 7 cannot divide 2, it must divide
as 7 and 2 are coprime. Note that 7 cannot divide
as square of a number always gives remainder 0, 1, 4, 2 when divided by 7 and never 6. But if
is divisible by 7 then
must give remainder 6 when divided by 7. Hence contradiction.
Necessary Lemma: square of a number always gives remainder 0, 1, 4, 2 when divided by 7
Key Ideas: Modular Arithmetic