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The function f is defined for each positive three-digit integer T by

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Bunuel
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The function f is defined for each positive three-digit integer T by [#permalink] New post   09 Apr 2021, 07:00
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The function f is defined for each positive three-digit integer T by \(f(T ) = 2^a3^b5^c\), where a, b and c are the hundreds, tens and units digits of T, respectively. If K and R are three-digit positive integers such that \(f(K) = 18f(R)\), then K − R =

(A) 65
(B) 70
(C) 80
(D) 100
(E) 120
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E
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The function f is defined for each positive three-digit integer T by [#permalink] New post   09 Apr 2021, 08:37
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Given: The function f is defined for each positive three-digit integer T by \(f(T ) = 2^a3^b5^c\), where a, b and c are the hundreds, tens and units digits of T, respectively.
Asked: If K and R are three-digit positive integers such that \(f(K) = 18f(R)\), then K − R =

Let K = abc where a= hundredth digit, b = tens digit & c= unit digit
and R = xyz where x= hundredth digit, y = tens digit & z= unit digit

\(f(K) = 2^a3^b5^c = 18* 2^x3^y5^z = 2*3^2*2^x3^y5^z = 2^{x+1}3^{y+2}5^z\)
a = x+1; b = y+2; c = z

K = 100a + 10b + c ; R = 100x + 10y + z
K-R = 100(a-x) + 10(b-y) + (c-z) = 100 + 20 + 0 = 120

IMO E
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Re: The function f is defined for each positive three-digit integer T by [#permalink] New post   11 Apr 2021, 08:39
Kinshook wrote:
Given: The function f is defined for each positive three-digit integer T by \(f(T ) = 2^a3^b5^c\), where a, b and c are the hundreds, tens and units digits of T, respectively.
Asked: If K and R are three-digit positive integers such that \(f(K) = 18f(R)\), then K − R =

Let K = abc where a= hundredth digit, b = tens digit & c= unit digit
and R = xyz where x= hundredth digit, y = tens digit & z= unit digit

\(f(K) = 2^a3^b5^c = 18* 2^x3^y5^z = 2*3^2*2^x3^y5^z = 2^{x+1}3^{y+2}5^z\)
a = x+1; b = y+2; c = z

K = 100a + 10b + c ; R = 100x + 10y + z
K-R = 100(a-x) + 10(b-y) + (c-z) = 100 + 20 + 0 = 120

IMO E


Kinshook why did you do a = x+1; b = y+2; c = z?
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Re: The function f is defined for each positive three-digit integer T by [#permalink] New post   11 Apr 2021, 08:44
hi CEdward
I have compared powers of 2, 3 and 5 which are prime numbers on both sides of the equation.

CEdward wrote:
Kinshook wrote:
Given: The function f is defined for each positive three-digit integer T by \(f(T ) = 2^a3^b5^c\), where a, b and c are the hundreds, tens and units digits of T, respectively.
Asked: If K and R are three-digit positive integers such that \(f(K) = 18f(R)\), then K − R =

Let K = abc where a= hundredth digit, b = tens digit & c= unit digit
and R = xyz where x= hundredth digit, y = tens digit & z= unit digit

\(f(K) = 2^a3^b5^c = 18* 2^x3^y5^z = 2*3^2*2^x3^y5^z = 2^{x+1}3^{y+2}5^z\)
a = x+1; b = y+2; c = z

K = 100a + 10b + c ; R = 100x + 10y + z
K-R = 100(a-x) + 10(b-y) + (c-z) = 100 + 20 + 0 = 120

IMO E


Kinshook why did you do a = x+1; b = y+2; c = z?
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The function f is defined for each positive three-digit integer T by [#permalink] New post   11 Apr 2021, 10:59
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Bunuel wrote:
The function f is defined for each positive three-digit integer T by \(f(T ) = 2^a3^b5^c\), where a, b and c are the hundreds, tens and units digits of T, respectively. If K and R are three-digit positive integers such that \(f(K) = 18f(R)\), then K − R =

(A) 65
(B) 70
(C) 80
(D) 100
(E) 120



suppose, \(a=7, b= 3, c = 1\)

\(f(R) = 2^73^35^1\) ;
\(R = 731\)

\(f(K) = 18*2^73^35^1 = 2*3^2* 2^73^35^1 = 2^83^55^1; \)
\(K = 851\)

\(K-R = 851 - 731 = 120\)

Answer E
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Re: The function f is defined for each positive three-digit integer T by [#permalink] New post   20 Apr 2024, 04:50
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Re: The function f is defined for each positive three-digit integer T by [#permalink]
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