Ones
Posted: February 27th, 2018, 12:36 pm
What is the smallest positive whole number with the property that when the digit 1 is appended to both ends, the new number is 99 times the original?
Cinelli
Cinelli
cinelli wrote:What is the smallest positive whole number with the property that when the digit 1 is appended to both ends, the new number is 99 times the original?
Cinelli
Spoiler:
Let our number be x. Then:
10x+10^n+1=99x
10^n+1=89x
where n is a whole number.
So some number starting with a 1, ending with a 1, and with just enough zeros to become exactly divisible by 89 is the one we want. An easy way to find the answer is to do a long division: start with 101 but add extra zeros repeatedly between the ones until 89 divides exactly. I won't do the working here, but the answer quickly drops out as:
x=112359550561797752809
GS