Need help plz. Don't really get this "reversing the digits" What is this really called in math language?
Prob:Find the product of 32 and 46. Now reverse the digits and find the product of 23 an 64. The products are the same. Does this happen with any pair of two digit numbers?
Find Two other pairs of two-digit numbers(with different digits)that have this property.
Explain your thinking in solving this problem and also include any mathematical work.
TY
No, it does not happen with any pair of 2-digit numbers, all we need is one counterexample,
e.g. 25 and 13
25x13 = 325
52x31 = 1612 , so not generally true
When is it true?
for the first number, let the tens digit be a, and the unit digit be b, thus the number is 10a + b
for the 2nd number, let the tens digit be c, and the unit digit be d, thus the number is 10c + d
product of pair
= (10a+b)(10c+d) = 100ac+10ad+10bc+bd
product of numbers reversed
= (10b+a)(10d+c) = 100bd+10bc+10ad+ac
we want
100ac+10ad+10bc+bd = 100bd+10bc+10ad+ac
which gives
99ac = 99bd
ac = bd
which shows that it must be true if the product of their tens digit equals the product of their units digits.
check for given pair
32 and 46
product of tens digit = 3x4 = 12
product of unit digits = 2x6 = 12
another pair would be
82 and 14
I will let you find a second example
Reiny, Thanks for helping to clarify and working out the problem.
The concept of reversing the digits of a number is often referred to as "reversing the order of the digits" in math language. This means that you flip the positions of the digits in the number.
To solve the given problem of finding two pairs of two-digit numbers that have the property of their products being the same when their digits are reversed, we can use a systematic approach.
Step 1: Find the product of 32 and 46.
32 multiplied by 46 equals 1,472.
Step 2: Reverse the digits of the product: 1,472 becomes 2,471.
Step 3: Find the product of 23 and 64.
23 multiplied by 64 also equals 1,472.
So we can see that the products of 32 and 46, and 23 and 64, are the same when their digits are reversed. This means that the given property holds true for this pair of two-digit numbers.
To find two other pairs of two-digit numbers with different digits that have the same property, we can use a similar approach.
Let's try the numbers 14 and 63.
14 multiplied by 63 equals 882.
Reversing the digits of 882 gives us 288.
Now, let's find the product of 41 and 36.
41 multiplied by 36 also equals 1,476.
Reversing the digits of 1,476 gives us 674.
Therefore, the products of 14 and 63, and 41 and 36, are the same when their digits are reversed.
In summary, we solved this problem by systematically multiplying two-digit numbers and then reversing the digits of the products. We found that the property of the products being the same when their digits are reversed holds true for the pairs 32 and 46, 23 and 64, 14 and 63, and 41 and 36.