The function $f$ is defined for all positive, three-digit integers as $f(x)=(2^a)(3^b)(5^c),$ where $a, b,$ and

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Vincen: Legendary Member; Posts: 2898; Joined: Thu Sep 07, 2017 2:49 pmFollowed by: 5 members; Thanked: 6 times

by Vincen » Fri Jul 10, 2020 2:54 am

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The function $f$ is defined for all positive, three-digit integers as $f(x)=(2^a)(3^b)(5^c),$ where $a, b,$ and $c$ are the hundreds, tens, and units digits, respectively, of $x.$ What is $m-n?$

(1) $f(m)=8f(n)$

(2) $n=111$

[spoiler]OA=A[/spoiler]

Source: Veritas Prep

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Source: Beat The GMAT — Data Sufficiency | Fri Jul 10, 2020 2:54 am

deloitte247: Legendary Member; Posts: 2214; Joined: Fri Mar 02, 2018 2:22 pmFollowed by: 5 members

Re: The function $f$ is defined for all positive, three-digit integers as $f(x)=(2^a)(3^b)(5^c),$ where $a, b,$ an

by deloitte247 » Fri Jul 17, 2020 4:59 am

$$Given\ that:\ f\left(x\right)=\left(2^a\right)\left(3^b\right)\left(5^c\right)$$
$$where\ a,\ b,\ and\ c\ =\ hundred,tens\ and\ unit\ digit\ of\ x$$
$$T\arg et\ question\ =>\ what\ is\ m-n?$$
$$Statement\ 1=>\ f\left(m\right)=8f\left(n\right)$$
$$f\left(m\right)=2^3f\left(n\right)$$
$$let\ f\left(n\right)=\left(2^x\right)\left(3^y\right)\left(5^z\right)$$
$$\ f\left(m\right)=2^3\cdot\left(2^x\right)\left(3^y\right)\left(5^z\right)$$
$$\ f\left(m\right)=2^{\left(3+x\right)}\left(3^y\right)\left(5^z\right)$$
$$m=\left[\left(3+x\right)\cdot100\right]+\left(y\cdot10\right)+\left(z\cdot1\right)$$
$$n=\left(x\cdot100\right)+\left(y\cdot10\right)+\left(z\cdot1\right)$$
$$m=300+100x+10y+z$$
$$n=100x+10y+z$$
$$m-n=300$$
$$Statement\ 1\ is\ SUFFICIENT$$
$$Statement\ 2=>n=111$$
$$f\left(n\right)=\left(2^1\right)\left(3^1\right)\left(5^1\right)$$
$$but\ there\ is\ no\ \inf ormation\ on\ the\ value\ of\ m\ or\ f\left(m\right)hence\ statement\ 2\ is\ NOT\ SUFFICIENT$$
$$\sin ce\ only\ statement\ 1\ is\ SUFFICIENT,$$
$$Answer\ =\ A$$

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