Is that Divisible?

Question text

A number 32a5b¯¯¯¯¯¯¯¯¯¯¯¯

is divisible by 11. Its last two digits form a number 5b¯¯¯¯¯ that can be exactly divided by 3. What is the largest 5

-digit number with this with these properties?

Your Answer:

Solution

Since 5b¯¯¯¯¯

is divisible by 3, b should be either 1, 4 or 7. If 32a5b¯¯¯¯¯¯¯¯¯¯¯¯ is to be divisible by 11, it must be the case that

3−2+a−5+b=a+b−4

is a multiple of 11. If b=1, then a=3; if b=4 then a=0; and if b=7, then a=8. The three 4-digit numbers that satisfy these properties are then 32351, 32054 and 32857. Thus, the largest 5-digit number with these properties is 32857

.

The correct answer is: 32857

 

My Solution (That above one was crappy)

 If 5b is divisible by 3, then b must be 1,4, or 7, because only for those values will 5+b be divisible by 3. If 32a5b is divisible by 11, we must have 3-2+a-5+b, or a+b-4, must be divisible by 11. Let's say b is 7. Then a has to be 8. So, the resulting number is 32857. But what if b is 4? Then a is 0, for the answer to be 32054. If b is 1, then a is 3, 32351. The largest is 32857.

Important:

Remember that abcdef is divisible by 11 ifandonlyif a-b+c-d+e is divisible by 11.