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# If p and q are positive integers, is 21^p/630^q a terminating

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Senior PS Moderator
Joined: 26 Feb 2016
Posts: 3334
Location: India
GPA: 3.12
If p and q are positive integers, is 21^p/630^q a terminating  [#permalink]

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31 Dec 2017, 23:02
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Difficulty:

75% (hard)

Question Stats:

20% (01:46) correct 80% (02:03) wrong based on 26 sessions

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If p and q are positive integers, is $$\frac{(21)^p}{(630)^q}$$ a terminating decimal

(1) p < 2q
(2) p > q

Source: Experts Global

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Joined: 07 Dec 2017
Posts: 868
Re: If p and q are positive integers, is 21^p/630^q a terminating  [#permalink]

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01 Jan 2018, 01:20
pushpitkc wrote:
If p and q are positive integers, is $$\frac{(21)^p}{(630)^q}$$ a terminating decimal

(1) p < 2q
(2) p > q

Source: Experts Global

We'll simplify the expression so we understand what we need to do.
This is a Precise approach.

We'll extract all common factors between 21 and 630, creating an easier-to-understand expression.
$$\frac{(21)^p}{(630)^q}=\frac{(21)^p}{(21*30)^q}=\frac{(21)^{p-q}}{(30)^q}=\frac{(3*7)^{p-q}}{(3*10)^q}=\frac{(3)^{p-2q}*(7)^{p-q}}{(10)^q}$$
A terminating decimal is an integer divided by some power of 10.
Therefore, for the above to be a terminating decimal, $$(3)^{p-2q}*(7)^{p-q}$$ must be an integer.
As 3 and 7 are both prime numbers then this occurs only when p-2q and p-q are both positive. This happens only when p>2q.
Looking at our statements (1) lets us answer the question with a NO and (2) does not let us answer the question.

*Note 1 - a terminating decimal is basically one long integer with the decimal point somewhere in the middle.
Therefore you can multiply it by a power of 10 to create an integer. This also means that every terminating decimal is an integer divided by a power of 10.
*Note 2 - A number divided by another number is an integer only if the denominator has the same prime factors as the numerator.
As this can never happen for powers of 3 and 7, they must both be in the numerator meaning that p-2q and p-q need to be positive.
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Re: If p and q are positive integers, is 21^p/630^q a terminating &nbs [#permalink] 01 Jan 2018, 01:20
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