Answers

2015-04-24T16:51:36+05:30

This Is a Certified Answer

×
Certified answers contain reliable, trustworthy information vouched for by a hand-picked team of experts. Brainly has millions of high quality answers, all of them carefully moderated by our most trusted community members, but certified answers are the finest of the finest.
Q1 :
Let P' be the number trucated after 1000 digits.

Since 16\,\mid\,10^4, it is sufficient to check the last four digits of P' :
1-9     : 1*9 = 9 digits
10-99   : 90*2 =180 digits
100-370 : 271*3= 813 digits

So P' ends in \ldots 3683693 and the last four digits are 3693
3693=16\times 230+\boxed{13}

So the remainder is 13 when P' is divided by 16.

Q2 :
The remainder will be same as the remainder when sum of digits of P is divided by 9(we don't need to worry about placevalue here as 10^n\equiv 1\pmod{9} for all n\in\mathbb{Z^+}) :
1+2+3+\cdots+1000=\dfrac{1000(1000+1)}{2}=500500\equiv 5+5=10\equiv\boxed{1}

Therefore the remainder is 1 when P is divided by 9.
1 3 1