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

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

Why can't z be 4?

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

If M is 329, value of z is 9. You're getting different values of z, making your solution incorrect.

I didn't understand. Can you please explain?

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

This is one of the best question I have seen in this forum... though this should have been a 95% difficulty question.
Thanks for the solution.

A good solution to get there in 2 minutes might look like this on paper:

M = xyz
M = y(10z + y)

y is prime
(10z + y) is prime
Note - the units digit of 10z + y is y.

From y being a prime digit, we know it must be in {2, 3, 5, 7}

For a multi-digit number to be prime, it's units digit must be in {1, 3, 7, 9}.
So given 10z + y is prime, whose units digit is y, we know that y must be in that set.

The intersection of these two constraints means that y must be in {3, 7}. Two cases to test, but how many permutations of z will we have to test? Hopefully not many.

Ah, another thing: the units digit of a product of two numbers is the product of their units digits. i.e. the units digit of ab * cd == units digit of b*d.
So given that M = y(10z+y), the units digits of M is the units digit of y*y.

Putting this together into a table of possibilities:

y	y*y	z
3	9	9
7	49	9

Great, only one option for z.

Testing these possibilities we find:

M = 3 * (10 * 9 + 3) = 279 // Doesn't fit, the y digit comes out as 7 instead of 3.
M = 7 * (10 * 9 + 7) = 679 // This works!