Solved Examples

[ravihanda]

a^2 + b^2 = 1553, a and b are integers, then (a+b) divided by 7 cannot have the remainder:
1)1 2)2 3)4 4)5 5)6
Try and break 1553. You will get a = +/- 32 and b = +/- 23 or vice versa.
So, a + b can take the following values -> 55, 9, -55, -9.
Remainders from 7 are -> 6, 2, -6, -2 which are as good as 6,2,1,5 respectively.
So, my answer is 4 (Option 3).

[gokulrndl@gmail.com]

Q. What is the remainder when 2^2+22^2+222^2.........+(222........49twos)^2 is divided by 9?

Rem of 2^2 is 4
Rem of 22^2 = 4^2 = 16 = 7
Rem of 222^2 = 6^2 = 36 = 0
Rem of 2222^2 = 8^2 = 64 = 1
Rem of 22222^2 = 10^2 = 1^2 = 1
Rem of 222222^2 = 12^2 = 3^2 = 0
Rem of 2222222^2 = 14^2 = 5^2 = 25 = 7
Rem of 22222222^2 = 16^2 = 7^2 = 49 = 4
Rem of 222222222^2 = 18^2 = 0
After this the pattern would repeat.
So, for the first 9 terms, remainder is = 4+7+0+1+1+0+7+4+0 = 24 = 6
There are 49 terms in total. For each 9 terms from the beginning the remainder is 6.
So, till 45 terms the remainder is 30 {5 sets of 9, each gives 6}
So, till 45 terms I can say that the remainder is 3
4 terms after that will give me 4,7,0 and 1.
Net remainder = 3 (from 45 terms) + 4 + 7 + 0 + 1 = 15 = 6
The answer is 6.

Can you please explain how

22^2 = 4^2
222^2 = 6^2 and so on.... please

[ravihanda]

The remainder of 22 from 9 is 4. So it can be replaced with 4 when finding out the remainder by 9.
The remainder of 222 from 9 is 6. So it can be replaced with 6 when finding out the remainder by 9.
And so on.....

[grimreaper]

Q. What is the remainder when 2^2+22^2+222^2.........+(222........49twos)^2 is divided by 9?

Rem of 2^2 is 4
Rem of 22^2 = 4^2 = 16 = 7
Rem of 222^2 = 6^2 = 36 = 0
Rem of 2222^2 = 8^2 = 64 = 1
Rem of 22222^2 = 10^2 = 1^2 = 1
Rem of 222222^2 = 12^2 = 3^2 = 0
Rem of 2222222^2 = 14^2 = 5^2 = 25 = 7
Rem of 22222222^2 = 16^2 = 7^2 = 49 = 4
Rem of 222222222^2 = 18^2 = 0
After this the pattern would repeat.
So, for the first 9 terms, remainder is = 4+7+0+1+1+0+7+4+0 = 24 = 6
There are 49 terms in total. For each 9 terms from the beginning the remainder is 6.
So, till 45 terms the remainder is 30 {5 sets of 9, each gives 6}
So, till 45 terms I can say that the remainder is 3
4 terms after that will give me 4,7,0 and 1.
Net remainder = 3 (from 45 terms) + 4 + 7 + 0 + 1 = 15 = 6
The answer is 6.

Can you please explain how

22^2 = 4^2
222^2 = 6^2 and so on.... please

its called the remainder chain........ 22 leaves remainder 4 with 9.. so reduce 22^2 as 4^2.. moreover... to find the remainder... see that... 2222....n times raise to power 2.... when divided by 9...is equivalent to (2Xn)^2
eg.. 22^2 = (2X2)^2 = 4^2

22222^2 = (2X5)^2=(10)^2... so on

so after 2...9times raised to power 2... the pattern repeats......

[gokulrndl@gmail.com]

The remainder of 22 from 9 is 4. So it can be replaced with 4 when finding out the remainder by 9.
The remainder of 222 from 9 is 6. So it can be replaced with 6 when finding out the remainder by 9.
And so on.....

Thanks a lot buddy.....

One more question: what is the remainder when 63! is divided by 71. I tried but didnt get the answer. I used euler theorm to find out but too too tedious. Can you please help with this one?

[grimreaper]

The remainder of 22 from 9 is 4. So it can be replaced with 4 when finding out the remainder by 9.
The remainder of 222 from 9 is 6. So it can be replaced with 6 when finding out the remainder by 9.
And so on.....

Thanks a lot buddy.....

One more question: what is the remainder when 63! is divided by 71. I tried but didnt get the answer. I used euler theorm to find out but too too tedious. Can you please help with this one?

for such Qs go backwards... as in.... start from 70!... once again use chain rule of remainders.... so now i knw 70!+1 is a mult of 71, since 71 is prime...
so 63!*64*65....70 gives remainder -1 with 71
64 is nothing but -7, same way 65 is -6.. so a product is got (-7*-6**-5....-1) =1
so 63! gives a remainder of -1 with 71
or tht is... remainder of 70

[gokulrndl@gmail.com]

The remainder of 22 from 9 is 4. So it can be replaced with 4 when finding out the remainder by 9.
The remainder of 222 from 9 is 6. So it can be replaced with 6 when finding out the remainder by 9.
And so on.....

Thanks a lot buddy.....

One more question: what is the remainder when 63! is divided by 71. I tried but didnt get the answer. I used euler theorm to find out but too too tedious. Can you please help with this one?

for such Qs go backwards... as in.... start from 70!... once again use chain rule of remainders.... so now i knw 70!+1 is a mult of 71, since 71 is prime...
so 63!*64*65....70 gives remainder -1 with 71
64 is nothing but -7, same way 65 is -6.. so a product is got (-7*-6**-5....-1) =1
so 63! gives a remainder of -1 with 71
or tht is... remainder of 70

Iam still not getting this steP:
64 is nothing but -7, same way 65 is -6.. so a product is got (-7*-6**-5....-1) =1
Can you please explain this.

[grimreaper]

Iam still not getting this steP:
64 is nothing but -7, same way 65 is -6.. so a product is got (-7*-6**-5....-1) =1
Can you please explain this.

64 gives a negative remainder of -7 with 71, same way 65 and so on.... so see, 7*6*5.... is nothing but 7! =5040 which leaves a remainder 70 when divided by 71... but we have one minus sign... so it becomes -70 which corresponds to 1
hope ya got it now

[gokulrndl@gmail.com]

Iam still not getting this steP:
64 is nothing but -7, same way 65 is -6.. so a product is got (-7*-6**-5....-1) =1
Can you please explain this.

64 gives a negative remainder of -7 with 71, same way 65 and so on.... so see, 7*6*5.... is nothing but 7! =5040 which leaves a remainder 70 when divided by 71... but we have one minus sign... so it becomes -70 which corresponds to 1
hope ya got it now

Yes i got it but what we need here is 63! / 71 ka remainder right?

You have found out 70!/71 the remainder is -1 and 64*65*66......70/71 the remainder is 1. But what about 63!/71 ka remainder?

[grimreaper]

Iam still not getting this steP:
64 is nothing but -7, same way 65 is -6.. so a product is got (-7*-6**-5....-1) =1
Can you please explain this.

64 gives a negative remainder of -7 with 71, same way 65 and so on.... so see, 7*6*5.... is nothing but 7! =5040 which leaves a remainder 70 when divided by 71... but we have one minus sign... so it becomes -70 which corresponds to 1
hope ya got it now

Yes i got it but what we need here is 63! / 71 ka remainder right?

You have found out 70!/71 the remainder is -1 and 64*65*66......70/71 the remainder is 1. But what about 63!/71 ka remainder?

hmmm......see i have found out remainder of 63! only......... wht i have done is........ i kept 63! as it is... and worked backwards from 70!....
so let 63! give remainder A
so since i knw 70! gives remainder of -1, so i also got remainder of 64*65*66...till70 as 1

so i write 70! as 63!*64 65 66 67....

after 63! the numbers give a remainder of 1.... while overall remainder required is -1
means 63! must give a remainder -1 with 71....
i hope u realise tht am using remainder chain here........

so you get A= -1
or in other owrds... 63! leaves a remainder of -1 with 71... thus answer being 70