CAT 2018 Question Paper | Quants Slot 2

CAT Questions

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 N^N = 2¹⁶⁰ and N² + 2^N is an integral multiple of 2^x, then the largest possible x is [TITA]

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Video Explanation

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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 N^N = 2¹⁶⁰ and N² + 2^N is an integral multiple of 2^x
Now we have to find the largest possible value for x
N^N = 2¹⁶⁰
We can rewrite this 2¹⁶⁰ as (2^p)^q
such that 2^p = 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 + 2^N ⟹ 32² + 2³² = 2^x × k (where k is the integral multiple)
N² + 2^N ⟹ 2¹⁰ + 2³² = 2^x × k since 32² = (2⁵)²
N² + 2^N ⟹ 2¹⁰ (1 + 2²²) = 2^x × 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
X_max = 10
The largest possible x is 10

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

Hence, the answer is 10