Joey and Chloe and their daughter Zoe all have the same birthday. Joey

Question

Joey and Chloe and their daughter Zoe all have the same birthday. Joey is 1 year older than Chloe, and Zoe is exactly 1 year old today. Today is the first of the 9 birthdays on which Chloe's age will be an integral multiple of Zoe's age. What will be the sum of the two digits of Joey's age the next time his age is a multiple of Zoe's age?

A. 7
B. 8
C. 9
D. 10
E. 11

A. 7
B. 8
C. 9
D. 10
E. 11

udaypratapsingh99 · Accepted Answer

If a number is a multiple of both Chloe's age and Zoe's age, then it is a multiple of their difference. Since the difference between their ages does not change, then that means the difference between their ages has 9 factors. Therefore, the difference between Chloe and Zoe's age is 36, so Chloe is 37, and Joey is 38. The common factor that will divide both of their ages is 37, so Joey will be 74. 7 + 4 = 11 (E)

ljw1101 · Answer

UdayPratapSingh99  Can you please elaborate more on how 9 factors result in difference of 36? Thank you!

udaypratapsingh99 · Answer

Before that, can you tell me what are those no.s which have 9 factors?

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ljw1101 · Answer

UdayPratapSingh99

9, 18, 27, 36,45,...

udaypratapsingh99 · Answer

Wrong. Read carefully, i mentioned nine factors not nine as factor

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ljw1101 · Answer

UdayPratapSingh99

Legend! Found my mistake.

It should only be 36 -> 1,2,3,4,6,9,12,18,36

Hence, one of the 9 birthdays would be translated to one of the 9 factors....

Thank you!

udaypratapsingh99 · Answer

Hehe
Great. Now You understand the whole problem???

And don't forget to give Kudos uf i was of any help. It keeps us motivated.

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nik021 · Answer

how can we check what all the numbers have 9 factors?

udaypratapsingh99 · Answer

A number with 9 factors will be of one of two forms:
1) x^8
2) (x^2)*(y^2)
 where x & y are distinct and greater than 1.

for the smallest such number, use x=2 and y=3. Therefore, the number = (2^2)*(3^2) = 4*9 = 36.

Adit_ · Answer

One of the hardest problems imo. Is this actually a GMAT problem seems more like a brain teaser?

Kinshook · Answer

Joey and Chloe and their daughter Zoe all have the same birthday. Joey is 1 year older than Chloe, and Zoe is exactly 1 year old today. Today is the first of the 9 birthdays on which Chloe's age will be an integral multiple of Zoe's age. What will be the sum of the two digits of Joey's age the next time his age is a multiple of Zoe's age?

A. 7
B. 8
C. 9
D. 10
E. 11

What will be the sum of the two digits of Joey's age the next time his age is a multiple of Zoe's age?

Let the age of Joey, Chloe and Zoe be j, c & z respectively

j = c + 1
z = 1

(c + k)/(1 + k) will be an integer for 9 values of k.
(c-1)/(1+k) will be an integer for 9 values of k

c-1 will be of the form p1^2*p2^2 or of the form p^8
c-1 = 2^2*3^2 = 36; c = 37
c-1 = 2^2*5^2 = 100 exceeds 2 digits; not feasible
c-1 = 2^8 = 256; exceeds 2 digits; not feasible

c = 37
j = c+1 = 38

(38+k)/(1+k) is an integer
37/(1+k) is an integer
k = 36;

38+k = 38+36 = 74
The sum of the two digits of Joey's age the next time his age is a multiple of Zoe's age = 7+4 = 11

IMO E

Joey and Chloe and their daughter Zoe all have the same birthday. Joey

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Joey and Chloe and their daughter Zoe all have the same birthday. Joey

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