- Whenever there appears any term of the type a3 + b3 + c3, do check for a + b + c being equal to zero. If a + b + c is indeed zero, then a3 + b3 + c3 = 3abc.
- The series 1, 3, 6, 10, 15 should immediately be recognized as series of sum of first n natural numbers.
- To form all natural numbers from 1 to N by adding any natural numbers, one would just need 1, 2, 4, 8, 16, 32, 64…2n, where 2n is the largest power of 2 smaller than or equal to N. E.g. What is the minimum number of weights needed to be able to measure all natural numbers weights till 80, if weights can be kept only on one pan of the balance. One would need weights 1, 2, 4, 8,…64.
- To form all natural numbers from 1 to N by adding or subtracting any natural numbers, one would just need 1, 3, 9, 27, 81, ..3n. Be careful of the largest number needed in this case. E.g. What is the minimum number of weights needed to be able to measure all natural numbers weights till 80, if weights can be kept on both pans of the balance? One would need weights 1, 3, 9, 27, 81.
- In questions of the type where certain flowers/sweets/points etc gets diminished and then increases and again diminishes and again increases…rather than forming an equation, see if one can work backwards if final quantity is given or else work with options. Also in such cases, if 1/3rd the objects are given away, work on the objects that are remaining i.e. 2/3rd to save time. If working with options, select options intelligently e.g. I pick 1/3rd of the chocolates in a bowl and then return 3, next I pick 1/5th of the chocolates and then return 5, next…What is the number of chocolates in the bowl initially? The initial number of chocolates has to be a multiple of 3. Also 2/3rd of the initial number of chocolates plus 3 should be divisible by 5. This should be enough to reduce the possible options to just about 2.
- Remember that a2, b2, c2 or |a|, |b|, |c| are either zero or positive quantity. Thus solution to a2 + b2 + c2 = 0 or |a| + |b| + |c| = 0 is a = b = c = 0.
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