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DIRECTIONS for the question: Solve the following question and mark the best possible option.
Find the smallest positive integer k such that k (33+43+53+63)=an for some positive integers a and n with n >1.
- a)2
- b)4
- c)3
- d)1
Correct answer is option 'C'. Can you explain this answer?
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Verified Answer
DIRECTIONS for the question:Solve the following question and mark the ...
► K(33+43+53+63)=an
► K×432=an
Do prime factorization of 432
► K×24×33=an
for K×24×33 to be perfect power the least value of K=3 so that it becomes 64.
► K×432=an
Do prime factorization of 432
► K×24×33=an
for K×24×33 to be perfect power the least value of K=3 so that it becomes 64.
Most Upvoted Answer
DIRECTIONS for the question:Solve the following question and mark the ...
To find the smallest positive integer k such that k (33 43 53 63) = an for some positive integers a and n with n < 20,="" we="" need="" to="" find="" the="" smallest="" value="" of="" k="" that="" makes="" the="" expression="" a="" perfect="" power="" of="" />
Let's evaluate the expression 33 43 53 63:
33 43 53 63 = 33 * 43 * 53 * 63
To simplify the expression, we can find the prime factorization of each number:
33 = 3 * 11
43 = 43
53 = 53
63 = 3 * 3 * 7
Now, let's rewrite the expression:
33 43 53 63 = (3 * 11) * 43 * 53 * (3 * 3 * 7)
Since we want the expression to be a perfect power of n, we need to pair up the prime factors in such a way that each pair has the same exponent. This will ensure that the expression can be expressed as an integer raised to the power of n.
Pairing up the prime factors:
(3 * 3) * (11) * (43) * (53) * (7)
Now we have two pairs of 3's, and all the other prime factors are on their own. To make the expression a perfect power of n, we need to expand the pairs of 3's by multiplying them together. Let's do that:
(3 * 3) * (11) * (43) * (53) * (7)
= 9 * 11 * 43 * 53 * 7
Now we can rewrite the expression as a perfect power of n:
(9 * 11 * 43 * 53 * 7) = a^n
To find the smallest positive integer k, we need to find the smallest value of a^n where n < 20.="" since="" the="" prime="" factors="" in="" the="" expression="" are="" all="" distinct,="" the="" smallest="" value="" of="" a^n="" will="" be="" achieved="" when="" a="9" and="" n="2" (to="" use="" the="" pair="" of="" />
Therefore, the smallest positive integer k is:
k = 9 * 11 * 43 * 53 * 7 = 9 * 2 * 11 * 43 * 53 * 7 = 3 * 2 * 11 * 43 * 53 * 7 = 2,375,314.
So the answer is option (B) 2,375,314.
Let's evaluate the expression 33 43 53 63:
33 43 53 63 = 33 * 43 * 53 * 63
To simplify the expression, we can find the prime factorization of each number:
33 = 3 * 11
43 = 43
53 = 53
63 = 3 * 3 * 7
Now, let's rewrite the expression:
33 43 53 63 = (3 * 11) * 43 * 53 * (3 * 3 * 7)
Since we want the expression to be a perfect power of n, we need to pair up the prime factors in such a way that each pair has the same exponent. This will ensure that the expression can be expressed as an integer raised to the power of n.
Pairing up the prime factors:
(3 * 3) * (11) * (43) * (53) * (7)
Now we have two pairs of 3's, and all the other prime factors are on their own. To make the expression a perfect power of n, we need to expand the pairs of 3's by multiplying them together. Let's do that:
(3 * 3) * (11) * (43) * (53) * (7)
= 9 * 11 * 43 * 53 * 7
Now we can rewrite the expression as a perfect power of n:
(9 * 11 * 43 * 53 * 7) = a^n
To find the smallest positive integer k, we need to find the smallest value of a^n where n < 20.="" since="" the="" prime="" factors="" in="" the="" expression="" are="" all="" distinct,="" the="" smallest="" value="" of="" a^n="" will="" be="" achieved="" when="" a="9" and="" n="2" (to="" use="" the="" pair="" of="" />
Therefore, the smallest positive integer k is:
k = 9 * 11 * 43 * 53 * 7 = 9 * 2 * 11 * 43 * 53 * 7 = 3 * 2 * 11 * 43 * 53 * 7 = 2,375,314.
So the answer is option (B) 2,375,314.
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DIRECTIONS for the question:Solve the following question and mark the best possible option.Find the smallest positive integer k such that k (33+43+53+63)=anfor some positive integers a and n with n >1.a)2b)4c)3d)1Correct answer is option 'C'. Can you explain this answer?
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DIRECTIONS for the question:Solve the following question and mark the best possible option.Find the smallest positive integer k such that k (33+43+53+63)=anfor some positive integers a and n with n >1.a)2b)4c)3d)1Correct answer is option 'C'. Can you explain this answer? for CAT 2024 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about DIRECTIONS for the question:Solve the following question and mark the best possible option.Find the smallest positive integer k such that k (33+43+53+63)=anfor some positive integers a and n with n >1.a)2b)4c)3d)1Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for DIRECTIONS for the question:Solve the following question and mark the best possible option.Find the smallest positive integer k such that k (33+43+53+63)=anfor some positive integers a and n with n >1.a)2b)4c)3d)1Correct answer is option 'C'. Can you explain this answer?.
DIRECTIONS for the question:Solve the following question and mark the best possible option.Find the smallest positive integer k such that k (33+43+53+63)=anfor some positive integers a and n with n >1.a)2b)4c)3d)1Correct answer is option 'C'. Can you explain this answer? for CAT 2024 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about DIRECTIONS for the question:Solve the following question and mark the best possible option.Find the smallest positive integer k such that k (33+43+53+63)=anfor some positive integers a and n with n >1.a)2b)4c)3d)1Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for DIRECTIONS for the question:Solve the following question and mark the best possible option.Find the smallest positive integer k such that k (33+43+53+63)=anfor some positive integers a and n with n >1.a)2b)4c)3d)1Correct answer is option 'C'. Can you explain this answer?.
Solutions for DIRECTIONS for the question:Solve the following question and mark the best possible option.Find the smallest positive integer k such that k (33+43+53+63)=anfor some positive integers a and n with n >1.a)2b)4c)3d)1Correct answer is option 'C'. Can you explain this answer? in English & in Hindi are available as part of our courses for CAT.
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