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Question 1091638: The product of two ,2-digit numbers is2117.the product of thier units digit is27 and that of ten digit14 .find the numbers.
Answer by KMST(5328)   (Show Source):
You can put this solution on YOUR website! What digits could be the tens digits?
The pairs of factors whose product is 14 are
 and  .
If the product of the tens digits is 14,
the tens digits must be 2 and 7,
because 14 is not a digit.
What digits could be the units digits?
The pairs of factors whose product is 27 are
 and  .
If the product of the units digits is 27,
the units digits must be 3 and 9,
because 27 is not a digit.
THE FIFTH GRADER APPROACH:
What products are possible with the tens digits and the units digits listed above?
The 2 as a tens digit could be paired with either 3 or 9 as a units digit,
so the products could be
 and  .
Using approximate numbers to estimate:
  .
and   .
If a fifth grader just needs a quick answer,
he/she should pick  and  as the two numbers.
If a fifth grader needs is told "show your work,"
he/she should just multiply to verify.
A VERY SMART FIFTH GRADER
would use some modular arithmetic (clock arithmetic) trick.
My grandma taught me a trick she called "nines check" when I was in second grade.
It involves adding the digits of numbers and any sum with more than one digit.
You can also ignore any 9 in your calculations.
You do it for factors and products, and then compare
the product of the factors' sums to
the sum of the product
to try to catch calculation mistakes.
In this case,the sums go like this:
For 2117, you can add 2+7=9, ignore the 9, and then add 1+1=2.
For 29 the sum is 2 (ignore the 0).
For 73 the first sum is 10, but you have to continue by adding  .
So, the sum for should be  ,
and than means that the "nines check" does not say that  is wrong.
and you know that is not  .
So, the "nines check" says that  is wrong.
On the other hand, the "nines check" does not say that is wrong.
For 23, the sum is 2+3=5.
For 79, the sum is 7 (ignore the 9).
 and the sum for 35 is 3+5=8,
which is not the same as 2, the sum for 2117 that we calculated before.
AN ALGEBRA STUDENT APPROACH
After finding the pairs of digits involved,
we know that the two numbers are  and  ,
where  and  are the units digits,
with  , meaning  or  .
The product of the two numbers is
 .
--> , too small a number.
--> --> ,
so the numbers are
 and 
Any way you get to the answer,
the two numbers are  and  .
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