Divisibilty + Remainders

Question 1

How to notice something is divisible by 4?

Answer 1

• Positive integer is a multiple of 4 when the integer formed by rightmost 2 digits is a multiple of 4, + not otherwise.

Question 2

How to notice something is divisible by 11?

Answer 2

-Positive integer is a multiple of 11 when “alternating sum” of its digits is a multiple of 11, EG: 86,691 is multiple of 11 as (8-6)+(6-9)+1=0x11
Eg: 2003-1505 = 498 NOT MULTIPLE of 11.
Eg; 10^7
1-0+0-0+0-0+0 (not a multiple of 11)

Question 3

What is the largest four-digit palindromic integer which is divisible by 15?

Answer 3

• For prime numbers p and q: an integer is divisible by pxq when it is both divisible by p and divisible by by q, and not otherwise.
• Positive integer is a number of 3 when digits add up to multiple of 3.
• positive integer is multiple of 5 when rightmost digit is either 0 or 5.

So answer 5775

Question 4

What is the remainder when the 2008-digit number 222…22 is divided by 9?

Answer 4

-when a positive integer is divided by 9, the remainder is the same as when the sum of digits is divided by 9.
2008x2= 4016
4016 / 9 = 446 r 2
Answer 2