What do you exactly do when you go about “solving" a problem ?
Consider this : Solving a problem only involves extracting pieces of information from the question at a deeper and deeper layer. While you do so, the mind peeps over this whole process and tries to find patterns to whatever data has been extracted as yet. And there comes a point – where you can – “pop" – just see the solution. The more you practice, the more skillful you become at the art of allowing the solution to “pop" up rather than rushing through, trying to “solve" the problem.
Confused? Let us see this process in action.
Consider the following problem :
Find the least number that gives a remainder of 9 when divided by 10, a remainder of 8 when divided by 9, a remainder of 7 when divided by 8 and so on … a remainder of 2 when divided by 1.
If you just get into solving the problem directly, it might turn out to be a real tough one. Many people get really clueless upon reading the above problem.
Let us see how we can extract lots of important pieces from the problem, till it finally crumbles itself.
Reconsider the problem. What does it mean ?
We are required to find a number N, such that,
N gives a remainder of 9 when divided by 10
N gives a remainder of 8 when divided by 9
N gives a remainder of 7 when divided by 8
N gives a remainder of 6 when divided by 7
…
…
N gives a remainder of 1 when divided by 2
Let this information sink in deeply before you proceed further. Is there a distinct pattern you notice here ?
Can you rephrase all the statements above so that a pattern is revealed ?
Here it is :
N gives a remainder of -1 when divided by 10
N gives a remainder of -1 when divided by 9
N gives a remainder of -1 when divided by 8
N gives a remainder of -1 when divided by 7
…
…
N gives a remainder of -1 when divided by 2
What next can you conclude from these statements ? What about considering (N + 1) ?
Right. (N + 1) gives a remainder of 0 when divided by 10, 9, 8, 7 … 2
i.e. (N + 1) is the least number which is a multiple of 10, 9, 8, 7, … 2.
Remember, we have done nothing to solve the problem as of yet. We are only rephrasing and playing around with the problem statements.
Yes, (N + 1) = L.C.M.(2,3,…10) = 2520
Hence, the required number is 2519.
Try this out everywhere. Instead of jumping over to “solving" a problem, PAUSE – spend more time with the problem statement, dig deeper, extract layers and layers of meaning from it, allow your mind to peep through all this and discover patterns. And pop will come the solution.
You cannot solve a problem. You can only allow a solution to emerge.
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