Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

If $x$ is the tens place digit and $y$ is the ones place digit of the product $725278\times 67066$, what is $x+y$? I have no idea how to even approach this.

share|improve this question
add comment

5 Answers

up vote 7 down vote accepted

Remember how you learned to multiply by hand?

       725278  
        67066  
       ------  
      .....68  
     ......8  
   .......      
  .......       
  -----------  
  .........48

This may help you see why, as others have pointed out, it’s enough to look at the last two digits of the two numbers.

share|improve this answer
add comment

It is enough to use the last $2$ digits:

Write $A=100a+b$ and $C=100c+d$, then $AC=10000ac+100(ad+bc)+bd$, all the rest is dividable by $100$, so would not affect the last $2$ digits.

In other words, using the notation $a\equiv b \pmod{100}$ for giving the same residue mod $100$, i.e. $100|a-b$, we have that $a\equiv b$ and $c\equiv d$ implies $ac\equiv bd$. In the giving example we have $$ 725278\equiv 78 \text{ and }67066\equiv 66 \pmod{100}.$$

share|improve this answer
2  
I don't quite understand why, could you elaborate further? –  MathScratch Dec 5 '12 at 23:44
add comment

So the product is $xy = 48641494348$, so $x = 4$ and $y = 8$.

However, as you see in the other answer, you don't actually have to compute the product. Note that $$ x = 725200 + 78\quad\text{and}\quad y = 67000 + 66 $$ so the product is $$\begin{align} (725200 + 78)(67000 + 66) &= 725200\times 67000 + 725200\times 66 + 67000\times 78 + 78\times 68 \\ &= Z + 78\times68. \end{align} $$ Here $Z$ is a number that has $0$ in the tens and ones places. So for the digits that you are looking for you need just consider the product $$ 78 \times 68 = 5148. $$ Again $x = 4$ and $y = 8$.

share|improve this answer
1  
thank you! This is perfect. –  MathScratch Dec 5 '12 at 23:49
add comment

The last two digits of $725278\times 67066$ are the last two digits of $78\times 66=5148$. Hence $x+y=4+8=12$.

share|improve this answer
add comment

66*78=5148

Therefore x=4 and y=8

share|improve this answer
    
Why can only looking at 66*78 work to solve this problem? –  MathScratch Dec 5 '12 at 23:43
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.