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11 Jun 2021, 07:45
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B
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
25%
(medium)
Question Stats:
75% (01:18) correct
25%
(01:40)
wrong
based on 63
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If \(125^{14}*48^8\) is written as an integer, how many consecutive zeroes will that integer have at the end?
(A) 22
(B) 32
(C) 42
(D) 50
(E) 112
(A) 22
(B) 32
(C) 42
(D) 50
(E) 112
11 Jun 2021, 07:56
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Bunuel
The answer would depend on the numbers of 2 or 5, whichever is less. WHY?
Because 10 is the product of 2 and 5, and lesser of two when multiplied by the same quantity of other would give you that many 10s.
The above will further get clarified by the prime factorisation.
\(125^{14}*48^8=(5^3)^{14}*(16*3)^8=5^{3*14}*2^{4*8}*3^8=5^{42}*2^{32}*3^8=(5*2)^{32}*5^{10}*3^8=10^{32}*5^{10}*3^8\)
So we have 10^32, which means 32 zeroes.
B
12 Jun 2021, 09:07
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To find the number of consecutive numbers we need to find what is the power of 10 in the given number.
And to find the power of 10 we need to find out how many 5's and 2's are there which can get multiplied to give us 10
\(125^{14}*48^8\)
125 can be written as \(5^3\)
=> \((5^3)^{14}*48^8\) (and 48 can be written as 16*3 )
=> \(5^{42}*(16*3)^8\) (16 can be written as \(2^4\))
=> \(5^{42}*(2^4*3)^8\)
=> \(5^{42}*2^{4*8}*3^8\)
=> \(5^{42}*2^{32}*3^8\)
=> \(5^{10}*5^{32}*2^{32}*3^8\)
=> \(5^{10}*10^{32}*3^8\)
So, we can have 32 consecutive 0's in the end as we have \(10^32\) as the maximum power of 10 in the number.
So, Answer will be B
Hope it helps!
And to find the power of 10 we need to find out how many 5's and 2's are there which can get multiplied to give us 10
\(125^{14}*48^8\)
125 can be written as \(5^3\)
=> \((5^3)^{14}*48^8\) (and 48 can be written as 16*3 )
=> \(5^{42}*(16*3)^8\) (16 can be written as \(2^4\))
=> \(5^{42}*(2^4*3)^8\)
=> \(5^{42}*2^{4*8}*3^8\)
=> \(5^{42}*2^{32}*3^8\)
=> \(5^{10}*5^{32}*2^{32}*3^8\)
=> \(5^{10}*10^{32}*3^8\)
So, we can have 32 consecutive 0's in the end as we have \(10^32\) as the maximum power of 10 in the number.
So, Answer will be B
Hope it helps!
