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abc is a three-digit number in which a is the hundreds digit
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12 Aug 2018, 15:52
12 Aug 2018, 15:52
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44% (01:24) correct
55% (01:29) wrong based on 88 sessions
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abc is a three-digit number in which a is the hundreds digit, b is the tens digit, and c is the units digit. Let \(&(abc)& = (2^a)(3^b)(5^c)\). For example, \(&(203)& = (2^2)(3^0)(5^3) = 500\). For how many three-digit numbers abc does the function &(abc)& yield a prime number?
(A) Zero
(B) One
(C) Two
(D) Three
(E) Nine
(A) Zero
(B) One
(C) Two
(D) Three
(E) Nine
Re: abc is a three-digit number in which a is the hundreds digit
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17 Aug 2018, 16:33
17 Aug 2018, 16:33
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Explanation
Since a prime number has only two factors, 1 and itself, \((2^a)(3^b)(5^c)\) cannot be prime unless the digits a, b, and c are such that two of the digits are 0 and the third is 1.
For instance, \((2^0)(3^1)(5^0) = (1)(3)(1) = 3\) is prime.
Thus, the only three values of abc that would result in a prime number & (abc)& are 100, 010, and 001. However, only one of those three numbers (100) is a three-digit number.
Since a prime number has only two factors, 1 and itself, \((2^a)(3^b)(5^c)\) cannot be prime unless the digits a, b, and c are such that two of the digits are 0 and the third is 1.
For instance, \((2^0)(3^1)(5^0) = (1)(3)(1) = 3\) is prime.
Thus, the only three values of abc that would result in a prime number & (abc)& are 100, 010, and 001. However, only one of those three numbers (100) is a three-digit number.
Re: abc is a three-digit number in which a is the hundreds digit
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Updated on: 04 Dec 2018, 18:54
Updated on: 04 Dec 2018, 18:54
Edit: Nevermind, silly mistake.
Originally posted by QuantumWonder on 02 Dec 2018, 08:50.
Last edited by QuantumWonder on 04 Dec 2018, 18:54, edited 2 times in total.
Last edited by QuantumWonder on 04 Dec 2018, 18:54, edited 2 times in total.
