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# abc is a three-digit number in which a is the hundreds digit

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abc is a three-digit number in which a is the hundreds digit [#permalink]  12 Aug 2018, 15:52
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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
[Reveal] Spoiler: OA

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Sandy
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GMAT Club Legend
Joined: 07 Jun 2014
Posts: 4749
GRE 1: Q167 V156
WE: Business Development (Energy and Utilities)
Followers: 93

Kudos [?]: 1655 [0], given: 396

Re: abc is a three-digit number in which a is the hundreds digit [#permalink]  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.
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Re: abc is a three-digit number in which a is the hundreds digit [#permalink]  02 Dec 2018, 08:50
Edit: Nevermind, silly mistake.

Last edited by QuantumWonder on 04 Dec 2018, 18:54, edited 2 times in total.
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Re: abc is a three-digit number in which a is the hundreds digit [#permalink]  02 Dec 2018, 19:33
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QuantumWonder wrote:
I am really confused. The function &abc& being 1,0,0 yields &100&=2^2*3^0*5^0 = 4. 4 is not a prime number. What am I misinterpreting?

hi
&abc& = $$2^a3^b5^c$$..
so here &abc& = &100&= $$2^13^05^0=2$$ and 2 is prime..
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Some useful Theory.
1. Arithmetic and Geometric progressions : https://greprepclub.com/forum/progressions-arithmetic-geometric-and-harmonic-11574.html#p27048
2. Effect of Arithmetic Operations on fraction : https://greprepclub.com/forum/effects-of-arithmetic-operations-on-fractions-11573.html?sid=d570445335a783891cd4d48a17db9825
3. Remainders : https://greprepclub.com/forum/remainders-what-you-should-know-11524.html
4. Number properties : https://greprepclub.com/forum/number-property-all-you-require-11518.html
5. Absolute Modulus and Inequalities : https://greprepclub.com/forum/absolute-modulus-a-better-understanding-11281.html

Re: abc is a three-digit number in which a is the hundreds digit   [#permalink] 02 Dec 2018, 19:33
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