Take any digit and multiply it by 239 and the product so obtained by 4649. what do you observe? explain your observation
Answers
So, you are taking some 3 digit number, say xyz, and then multiplying it by 239 and further multiplying it by 4649. The result is pretty much predictable and one could write it by virtue of just xyz being known, sans any actual multiplications by 239 and 4649.
For this, we will have to understand as to what exactly is happening as we multiply the given number by 239 and then by 4649 as a test case.
We will consider a simple case to start with.
Let the 3 digit number be 123. When we multiply it by 239, we get 29,397 and further multiplying it by 4649, we get 13,66,66,653. We do observe certain pattern but the entire logic is not very clear. Let’s multiply 239 by 4649 and voila, the number is 11,11,111.
You probably know the technique when we multiply any number by 11. If the number is say 47, 47*11 = 517. In here, we write the last digit as is, which happens to be 7. The next digit is written by adding 4 and 7 to give you 11. From 11 we pick the last digit as 1 and keep the carry 1. The number so far (in the answer of 47*11) is 17 and then we add the carry 1 to 4 to give us 5 and thus the final number is 517.
You can try it on any other number, say 1476. 1476*11 = 16236.
You place 6 as is the 7+6 is 13, place 3 and carry 1, 4+7 = 11+1(carry) = 12, place 2 and carry 1, 1+4 = 5+1 (carry) = 6, place 6 and there’s no carry and finally place 1, thus the product is 16236.
We have to do similarly, as we multiply any 3 digit number by 11,11,111 (as 239*4649 = 1111111).
Let the 3 digit number be 423. 423 * 1111111 = 46,99,99,953. The logic goes like this: You place the last digit 3 (from 423) as is and then you place the sum of 2+3=5 before 3 to get 53. Consider that you have consumed 1 and 1 out of 1111111 and you are left with 11111. For each of this digit, place the sum of 4+2+3 = 9 before 53 to obtain 99,99,953. Just like you added 2+3, now you add 4+2=6 and place it before 99,99,953 to obtain 6,99,99,953 and finally place 4 as is to obtain the final result as 46,99,99,953.
The logic is quite straightforward, once you understand it completely. Try it out on any number, you don’t really require a 3 digit number to start with. It could be a any number and you just need to master the technique of multiplying any number by a number consisting of n number of 1s.
If the technique is confusing, read it slowly and simultaneously practice it on a piece of paper. In the end, just verify your answers using a calculator until you gain enough confidence, so as not to require any further verification!