A boy claims that he can multiply any three-digit number by 1001 instantly. If his classmate says to him "715" he gives the answer immediately. Compute this answer and explain the boy's secret.
Ok, so the answer is obviously 715715. Also, the boy knows that his answer will always be in the hundred thousands value because :
1001*100=100100 (The lowest number having three digits)
1001*999=999999(The highest number having three digits)
Knowing that the value won't be one time in the thousands, another time in the hundreds,etc And knowing that its magnitude will always be in the hundred thousands, his trick is for sure to always work and won't have to worry about it changing. Now, as we can see, a certain pattern keeps showing up. The three digits number will always compose two times your answer.
So the boy can claim with insurance what the answer is without calculating anything. Now, why does the trick work ?
Let's work with 1001*715.
We can decompose 715 as : 700+10+5.
First, let's consider it this way:
Now we place them that way :
Ok, so we know that a certain pattern keeps repeating, that is, to show TWO times the three digits number in our answer. Now, what I did earlier was to consider the 7 and 1 as a unit to show clearly what happens when we add their true value to these two digits (1*10=10 and 7*100=700)
If we want the trick to work, we must assure ourselves that only one of our digits (7,1 or 5) will compose each column of the decimal place.(Which will also include 0's, which changes nothing)
7007 Now, should we should by *100, because we need our numbers (the two 7's) to be moved two times to meet our criterias
Exactly what we wanted. Lastly, why in our number 715715 why should we get a 715 (the one in darker color) and not zero's or any other number? That's because of the "1" (unit) in the number 1001
715000 (we are missing the other 715)
Here's what's happening :
And that's why the trick works. Did I answer the question correctly ? Or was I not hitting the core of the "trick" ?
Thank you !