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# A is a three digit number such that z is the units digit, y is the ten

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VP
Joined: 19 Oct 2018
Posts: 1312
Location: India
A is a three digit number such that z is the units digit, y is the ten  [#permalink]

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01 Sep 2019, 17:07
1
6
00:00

Difficulty:

85% (hard)

Question Stats:

39% (03:10) correct 61% (03:18) wrong based on 28 sessions

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M is a three digit number such that z is the units digit, y is the tens digit and x is the hundreds digit of M.
Also, M= y*(10z+y), where y and 10z+y are prime numbers. Find x+y+z ?

A. 12
B. 14
C. 17
D. 18
E. 22

Kudos for a correct solution.
Manager
Joined: 20 Jul 2019
Posts: 55
Re: A is a three digit number such that z is the units digit, y is the ten  [#permalink]

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01 Sep 2019, 18:57
1
3 digit number xyz can be written as xyz=100x+10y+z ,given y is prime so the max value of y is 7 ,now plug in values for z which ranges from 1to 9 considering y= 7 ,for z start from the bottom,I am substituting 9, now z= 10*9+7=97,given m=y*(10z+y)= 7 *97= 679 , Sox+y+z= 6+7+9= 22, the answer is E

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Intern
Joined: 08 Jul 2019
Posts: 16
Location: India
Schools: Copenhagen
Re: A is a three digit number such that z is the units digit, y is the ten  [#permalink]

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04 Sep 2019, 01:53
1
y is a prime, so possible values of y: 2,3,5,7,9
but (10z+y) is also a prime number number; so possible values of y: 3,7,9 (22,25,32,35 etc can never be prime).
Starting from maximum 2 digit prime number (we can start from min i.e. 13 as well, but we know 13*3 will never be a 3 digit number. we will have to check a lot of values to find first 3 digit number, so we start from back) => 10z+y = 97; 97*7=679 => 6+7+9 = 22

Thus E
VP
Joined: 19 Oct 2018
Posts: 1312
Location: India
Re: A is a three digit number such that z is the units digit, y is the ten  [#permalink]

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04 Sep 2019, 02:16
3
You are very close to the perfect solution. Though you get the correct answer, there are couple of mistakes in your solution.

1. 9 is not a prime number.
2. We don't have to look for a lot of values.

As y is a prime number, y can be 2,3,5 or 7
When y=2 or 5, 10z+y can never be prime.
Hence y can be 3 or 7

M=100x+10y+z=y*(10z+y)
$$100x+10y+z=10yz+y^2$$

Hence unit digit of $$y^2$$ is equal to z
z=unit digit of $$3^2$$ or $$7^2$$
z=9 in both cases

1. when y=3, 10z+y=93, which is not a prime
2. when y=7, 10z+y=97, which is a prime

M= 7*97=679

jimar wrote:
y is a prime, so possible values of y: 2,3,5,7,9
but (10z+y) is also a prime number number; so possible values of y: 3,7,9 (22,25,32,35 etc can never be prime).
Starting from maximum 2 digit prime number (we can start from min i.e. 13 as well, but we know 13*3 will never be a 3 digit number. we will have to check a lot of values to find first 3 digit number, so we start from back) => 10z+y = 97; 97*7=679 => 6+7+9 = 22

Thus E
Intern
Joined: 24 Jun 2019
Posts: 13
Re: A is a three digit number such that z is the units digit, y is the ten  [#permalink]

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01 Dec 2019, 09:23
nick1816 wrote:
M is a three digit number such that z is the units digit, y is the tens digit and x is the hundreds digit of M.
Also, M= y*(10z+y), where y and 10z+y are prime numbers. Find x+y+z ?

A. 12
B. 14
C. 17
D. 18
E. 22

Kudos for a correct solution.

Why can't z be 4?

then We can have 7*47 = 329 = 3+2+9 = 14. What is wrong with this solution?
VP
Joined: 19 Oct 2018
Posts: 1312
Location: India
Re: A is a three digit number such that z is the units digit, y is the ten  [#permalink]

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01 Dec 2019, 09:48
If M is 329, value of z is 9. You're getting different values of z, making your solution incorrect.

Krish728 wrote:
nick1816 wrote:
M is a three digit number such that z is the units digit, y is the tens digit and x is the hundreds digit of M.
Also, M= y*(10z+y), where y and 10z+y are prime numbers. Find x+y+z ?

A. 12
B. 14
C. 17
D. 18
E. 22

Kudos for a correct solution.

Why can't z be 4?

then We can have 7*47 = 329 = 3+2+9 = 14. What is wrong with this solution?
Intern
Joined: 24 Jun 2019
Posts: 13
Re: A is a three digit number such that z is the units digit, y is the ten  [#permalink]

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01 Dec 2019, 10:28
nick1816 wrote:
If M is 329, value of z is 9. You're getting different values of z, making your solution incorrect.

Krish728 wrote:
nick1816 wrote:
M is a three digit number such that z is the units digit, y is the tens digit and x is the hundreds digit of M.
Also, M= y*(10z+y), where y and 10z+y are prime numbers. Find x+y+z ?

A. 12
B. 14
C. 17
D. 18
E. 22

Kudos for a correct solution.

Why can't z be 4?

then We can have 7*47 = 329 = 3+2+9 = 14. What is wrong with this solution?

I didn't understand. Can you please explain?
VP
Joined: 19 Oct 2018
Posts: 1312
Location: India
Re: A is a three digit number such that z is the units digit, y is the ten  [#permalink]

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01 Dec 2019, 11:23
Krish728 Bro you are considering value of z=4

So value of M must be 'xy4', as M='xyz'

But, you are getting value of M= 329(value of z is 9); hence, you are contradicting your initial consideration
Re: A is a three digit number such that z is the units digit, y is the ten   [#permalink] 01 Dec 2019, 11:23
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