Dear All,

Pleas post your doubts of number theory sheet 3 on this thread.

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Dear All,

Pleas post your doubts of number theory sheet 3 on this thread.

Pleas post your doubts of number theory sheet 3 on this thread.

- sadmin
- Site Admin
**Posts:**228**Joined:**Wed Feb 13, 2013 5:51 am

Q23

301 = 7*43,

so 300 ≡ -1 (mod 7)

Then 300^3000 - 1 ≡ (-1)^3000 - 1 ≡ 1 - 1 ≡ 0 (mod 7)

So 7 divides 300^3000 - 1

297 = 27*11,

so 300 ≡ 3 (mod 11)

Then,

300^3000 - 1 ≡ 3^3000 - 1 ≡ (3^5)^600 - 1 (mod 11)

But 3^5 = 243 = 22*11 + 1

so 3^5 ≡ 1 (mod 11)

Then

300^3000 - 1 ≡ (3^5)^600 - 1 ≡ 1^600 - 1 ≡ 0 (mod 11)

So 11 divides 300^3000 - 1

Finally, 299 = 23*13,

so 300 ≡ 1 (mod 13)

Then

300^3000 - 1 ≡ 1^3000 - 1 ≡ 0 (mod 13)

So 13 divides 300^3000 - 1

Since 7, 11, 13 are all prime, it follows that their product, 1001 divides 300^3000 - 1

301 = 7*43,

so 300 ≡ -1 (mod 7)

Then 300^3000 - 1 ≡ (-1)^3000 - 1 ≡ 1 - 1 ≡ 0 (mod 7)

So 7 divides 300^3000 - 1

297 = 27*11,

so 300 ≡ 3 (mod 11)

Then,

300^3000 - 1 ≡ 3^3000 - 1 ≡ (3^5)^600 - 1 (mod 11)

But 3^5 = 243 = 22*11 + 1

so 3^5 ≡ 1 (mod 11)

Then

300^3000 - 1 ≡ (3^5)^600 - 1 ≡ 1^600 - 1 ≡ 0 (mod 11)

So 11 divides 300^3000 - 1

Finally, 299 = 23*13,

so 300 ≡ 1 (mod 13)

Then

300^3000 - 1 ≡ 1^3000 - 1 ≡ 0 (mod 13)

So 13 divides 300^3000 - 1

Since 7, 11, 13 are all prime, it follows that their product, 1001 divides 300^3000 - 1

- sadmin
- Site Admin
**Posts:**228**Joined:**Wed Feb 13, 2013 5:51 am

You are right - for both these questions it is better to approach from the options.

- sadmin
- Site Admin
**Posts:**228**Joined:**Wed Feb 13, 2013 5:51 am

Kindly solve question number 16 and 23 in the practice questions. Are we to proceed by options in both the questions or is there a shorter way out?

- gaurav1399
**Posts:**48**Joined:**Wed Feb 27, 2013 7:49 am

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