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The Algebraic Exponents is one of the most commonly tested topics in CAT exam. Questions like this have appeared consistently in the CAT exam for the last several years. To know more about the pattern of the CAT Exam and what type of questions to expect visit 2IIM's CAT Blueprint. The topic in the question below is very interesting as it is dwells on mere numbers. Try this question yourself before moving onto the solution provided.

Question 7 : If N and x are positive integers such that NN = 2160 and N2 + 2N is an integral multiple of 2x, then the largest possible x is [TITA]


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Explanatory Answer

Method of solving this CAT Question from Number Theory

Given that N and X are positive integers such that NN = 2160 and N2 + 2N is an integral multiple of 2x
Now we have to find the largest possible value for x
NN = 2160
We can rewrite this 2160 as (2p)q
such that 2p = N = q
pq = 160

We can try by substituting p and q with 2 and 80 respectively
(22)80 = 480
This doesn’t work so we can try by substituting p and q with 4 and 40 respectively
(24)40 = 1640
This also doesn’t work so now we can substitute p and q with 5 and 32 respectively
Such that (25)32 = 3232

Now we can say that N = 32
N2 + 2N ⟹ 322 + 232 = 2x × k (where k is the integral multiple)
N2 + 2N ⟹ 210 + 232 = 2x × k since 322 = (25)2
N2 + 2N ⟹ 210 (1 + 222) = 2x × k
This (1 + 222) is an odd number so the number is going to be 210 × multiple of the odd number
Hence the largest power of 2 can be 10 i.e. 210
Xmax = 10
The largest possible x is 10

The question is "If N and x are positive integers such that NN = 2160 and N2 + 2N is an integral multiple of 2x, then the largest possible x is [TITA]"

Hence, the answer is 10

 

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