Question

What is the remainder when 888 raised to 222 +222raised to 888
is divided by 5?​

Answer 1

Step-by-step explanation:

Method 1:

888222+222888≡3222+2888mod5

Next, we need work out the period of 3 and 2 mod 5.

34≡1mod5 and 24≡1mod5

So, the period for both are 4.

888222+222888≡3222+2888≡34×55+2+24×222≡32+1≡0mod5

So, the answer is 0.

Method 2:Since we are only dealing with division by 5, all we need to care is the last digit.

Keep multiply 8 by itself and we get 8, 64, 512, 4096,… Just looking at the last digit, we get 8,4,2,6,8,4,2,6,… and we can see that the pattern repeats every 4 terms. So, in order to know what is the last digit of 888222 , all we need is take 222mod4 and take the corresponding term. The last digit is 4 in this case.

The pattern for 2 will be 2,4,8,6,2,4,8,6,… and the pattern also repeats every 4 terms. Since 888 is a multiple of 4, we know that 222888 has a last digit of 6.

Add those 2 up and we get last digit to be 0 and we show that the number is divisible by 5 thus remainder is 0.