Such a hard work for me to solve this problem. There're some techniques that I will use to solve this:

xxxab * 11 = xxxxcb ( where c is the unit digit of a+b)

For example 23*11 = 253; note that 5 = 2+3
or 478* 11 = xx58 ( where 5 is the unit digit of 7 + 8)

(a+b)(a-b) = a^2 - b^2

the product of xxxab * xxxcd wil have the same unit and tenth digit with that of ab*cd
For example : 1114 * 5678 = xxxx92 and 14*78 = xx92

xxx5 * xxx5 = xxxxx25

First, I devide them into 3 pairs:
22x31
44x27
37x43

22 x 31 = 2*11* 31 = 2* xx41 ( use rule 1 when multiply with 11)
= xx82 = 82 (rule 3)

44x27 = 4*11*27 = 4*xx97 = 4*97 = 4*(100-3) = 400 - 12 = 388 = 88

37x43 = (40-3)*(40+3) = 40^2 - 3^2 = 1600 - 9 = 1591 = 91

Now we have 82*88*91

82*88 = (85-3)(85+3) = 85^2 - 3^2 = xx25 - 9 = xx16 = 16

Now we have only 16*91 left

16*91 = 16*(100-10 +1) = 1600 - 160 +16 = xx40 + 16 = xx56 = 56

Then Pick D.

I want to know if anyone has a shorter approach. It took me more than 5 minutes to solve this.

Another solution I have thought of is :

xxxab* xxxcd = xxxxxxxef

Where ab*d + b*c*10 = xef

is it gmat question?

Answer: D

Let´s try to avoid "brutal force", be quick and think about something "thinkable" during GMAT test conditions...

37.43 = (40-3)(40+3) = 40^2 - 9 = final 91 very easy

44.27 = (40+4)(30-3) = (distributive gets) final equal to 100-12 = final 88 easy

22.31 = (20+2)(30+1) = final 82 very easy

Now the answer is the last two digits of 91.88.82

Observing that 88.92 = (85+3)(85-3) = (distributive gets) final 25-9 = final 16 easy

Finally 91.16 = (20-4)(90+1) = (distributive gets) 1800-344 = final 56 easy.

(There must be even more quick paths, but this one should come to the answer in approx. 2min, therefore it is good enough for GMAT purposes...)

Hope you like it!

limestone wrote:Such a hard work for me to solve this problem. There're some techniques that I will use to solve this:

xxxab * 11 = xxxxcb ( where c is the unit digit of a+b)

For example 23*11 = 253; note that 5 = 2+3
or 478* 11 = xx58 ( where 5 is the unit digit of 7 + 8)

(a+b)(a-b) = a^2 - b^2

the product of xxxab * xxxcd wil have the same unit and tenth digit with that of ab*cd
For example : 1114 * 5678 = xxxx92 and 14*78 = xx92

xxx5 * xxx5 = xxxxx25

First, I devide them into 3 pairs:
22x31
44x27
37x43

22 x 31 = 2*11* 31 = 2* xx41 ( use rule 1 when multiply with 11)
= xx82 = 82 (rule 3)

44x27 = 4*11*27 = 4*xx97 = 4*97 = 4*(100-3) = 400 - 12 = 388 = 88

37x43 = (40-3)*(40+3) = 40^2 - 3^2 = 1600 - 9 = 1591 = 91

Now we have 82*88*91

82*88 = (85-3)(85+3) = 85^2 - 3^2 = xx25 - 9 = xx16 = 16

Now we have only 16*91 left

16*91 = 16*(100-10 +1) = 1600 - 160 +16 = xx40 + 16 = xx56 = 56

Then Pick D.

I want to know if anyone has a shorter approach. It took me more than 5 minutes to solve this.

Another solution I have thought of is :

xxxab* xxxcd = xxxxxxxef

Where ab*d + b*c*10 = xef

Hats off to such a hard work, limestone. Actually, the number formed by the last two digits to the right is same as the remainder when the product is divided by 100.

The remainder when 22 X 31 X 44 X 27 X 37 X 43 is divided by 100

= The remainder when 22 X 31 X 11 X 27 X 37 X 43 is divided by 25 {this division by 4 should be compensated in the end}

= The remainder when 22 X 6 X 11 X 2 X 12 X 18 is divided by 25

= The remainder when 132 X 22 X 216 is divided by 25

= The remainder when 7 X 22 X 16 is divided by 25

= The remainder when 154 X 16 is divided by 25

= The remainder when 4 X 16 or 64 is divided by 25

= 14 {don't forget to multiply this by 4}

Hence, the required two-digit number is [spoiler]56.


D[/spoiler]

Very nice approaches, buddies. That will help me find out the answer within 2 minutes in a real test. By the way, where did you get such a good method to solve this, sanju?
Great job and keep posting more interesting questions.