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# How many positive even integers are factors of 72ˆ3?

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Intern
Joined: 07 Oct 2018
Posts: 1
How many positive even integers are factors of 72ˆ3?  [#permalink]

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13 Oct 2019, 13:38
1
6
00:00

Difficulty:

65% (hard)

Question Stats:

49% (01:46) correct 51% (01:51) wrong based on 76 sessions

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How many positive even integers are factors of 72ˆ3?

a. 9
b. 10
c. 31
d. 63
e. 67
Math Expert
Joined: 02 Aug 2009
Posts: 8187
Re: How many positive even integers are factors of 72ˆ3?  [#permalink]

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01 Nov 2019, 20:07
1
1
cinthiaph wrote:
How many positive even integers are factors of 72ˆ3?

a. 9
b. 10
c. 31
d. 63
e. 67

Remember, it is always better to find Odd factors and then subtract from total factors....
$$72^3=(2^33^2)^3=2^93^6$$

(I) Total factors ----- $$(9+1)(6+1)=10*7=70$$

(II) Odd factors -- take just the odd prime numbers =$$3^6.....(6+1)=7$$

(III) Even factors -- Total - odd = $$70-7=63$$

D
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VP
Joined: 19 Oct 2018
Posts: 1078
Location: India
Re: How many positive even integers are factors of 72ˆ3?  [#permalink]

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13 Oct 2019, 14:08
1
$$72^3= 2^9*3^6$$
Factors of 72*3 can be written as $$2^a*3^b$$

If factors are even, a can take 9 values (1 to 9) and b can take 7 values (0 to 6)
Total positive even factors= 9*7=63

cinthiaph wrote:
How many positive even integers are factors of 72ˆ3?

a. 9
b. 10
c. 31
d. 63
e. 67
SVP
Joined: 26 Mar 2013
Posts: 2341
How many positive even integers are factors of 72ˆ3?  [#permalink]

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13 Oct 2019, 14:28
2
cinthiaph wrote:
How many positive even integers are factors of 72ˆ3?

a. 9
b. 10
c. 31
d. 63
e. 67

$$72^3= 2^9*3^6$$
Then total factors = (9+1) * (6+1) = 70

Odd factors = (6+1) =7

even factors = 70 - 7 = 63

Senior Manager
Joined: 10 Apr 2018
Posts: 279
Location: India
Concentration: General Management, Operations
GMAT 1: 680 Q48 V34
GPA: 3.3
How many positive even integers are factors of 72ˆ3?  [#permalink]

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14 Oct 2019, 05:09
1
1
$$72^3=2^9∗3^6$$
Thus, total number of factors= (9+1)*(6+1)=70
So, number of odd factors=(6+1)=7
Therefore, number of even factors=70-7=63
Intern
Joined: 17 Oct 2019
Posts: 2
Re: How many positive even integers are factors of 72ˆ3?  [#permalink]

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03 Nov 2019, 11:28
(9+1) (6+1) i dont get it

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Re: How many positive even integers are factors of 72ˆ3?   [#permalink] 03 Nov 2019, 11:28
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