Z is a two digit positive integer, Z is equal to 7 times the sum of it

Question

Z is a two digit positive integer, Z is equal to 7 times the sum of its digits. Is Z = 84?

(1) Sum of its digits is 12
(2) Product of its digits is 32

Bunuel · Answer

Let the tenths and units digits of Z be  a  and  b  respectively, so  Z=10a+b .

Given:  Z=10a+b=7(a+b)  -->  a=2b  -->  Z=10a+b=20b+b=21b , so Z could take only the following 4 values: 21, 42, 63, and 84.

Question: is  Z=84 ?

(1) Sum of its digits is 12 --> from the possible values of Z only 84 has the sum of its digits equal to 12. Sufficient.
(2) Product of its digits is 32 --> from the possible values of Z only 84 has the product of its digits equal to 32. Sufficient.

Answer: D.

Creeper300 · Answer

good points, the answer should be D. Since 4,8 are the only options. and only 84 is seven times the sum.

chaoswithin · Answer

Wow, thank you Bunuel, I would have totally gotten this wrong on the test haha.

Better here than during the real GMAT.

srjaidev · Answer

let Z = 10a + b, where a is the tens place digit and b is the ones place digit

10a + b = 7 (a+b) => a = 2b ----- (0)

(1) states that the (a+b) = 12
using (1) and (0), the only possible answer is 84

(2) states that axb = 32
using (2) and (0), the only possible answer is 84.

gtr022001 · Answer

my brain is not clicking, but can someone explain how a = 2b?

i'm understanding this part: 10a + b = 7 (a+b)

chaoswithin · Answer

Well 10a + b is just another way of writing a two digits number.

For example, if Z = 65, then Z = 65 = 10*6 + 5. where 6 = "a" and 5 = "b"
taking things further, we can express any length number using the notation above.

so if Z = 12345.6 = 10000*1 + 1000*2 + 100*3 + 10*4 + 5 + 0.1*6

and since we are given the fact that Z = 7 * (sum of the digits) = 7* (a+b)
we are able state that 10a + b = 7 (a+b).

Bunuel · Answer

10a + b = 7 (a+b)  -->  10a+b=7a+7b  -->  3a=6b  -->  a=2b .

Hope it's clear.

gtr022001 · Answer

ah, crystal clear, thanks guys!

prashantbacchewar · Answer

Thanks for post.. Nice question

bumpbot · Answer

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Z is a two digit positive integer, Z is equal to 7 times the sum of it

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