# While adding ten two-digit numbers the digits of one of the numbers were interchanged. As a result the sum of all the ten numbers increased by a value which was four less than that number. Three times the sum of the digits of the original number is ten less than the number. What is the product of the digits of that number?

Please help how to obtain that number. and also explain (While adding ten two-digit numbers the digits of one of the numbers were interchanged. As a result the sum of all the ten numbers increased by a value which was four less than that number ) this sentence

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1. a1 ÷ a10 = your numbers.

You can write 10th digit in form:

a10 = 10 x + y

Original sum:

S = a1 + a2 +...+ a9 + 10 x + y

If 10th digit be interchanged new number will be 10 y + x

New sum wil be:

S1 = a1 + a2 +...+ a9 + 10 y + x

The sum of all the ten numbers increased by a value which was four less than that number mean:

S1 - S = 10 x + y - 4

S1 - S = a1 + a2 +...+ a9 + 10 y + x - ( a1 + a2 +...+ a9 + 10 x + y )

S1 - S = ( a1 + a2 +...+ a9 ) + 10 y + x - ( a1 + a2 +...+ a9 ) - 10 x - y

S1 - S = 10 y + x - 10 y + x

S1 - S = 9 y - 9 x

So:

S1 - S = 10 x + y - 4

9 y - 9 x = 10 x + y - 4

9 y - y = 10 x + 9 x - 4

8 y = 19 x - 4

Three times the sum of the digits of the original number is ten less than the number mean:

3 ( x + y ) = 10 x + y - 10

3 x + 3 y = 10 x + y - 10

3 y - y = 10 x - 3 x - 10

2 y = 7 x - 10

Now you must solve system:

8 y = 19 x - 4

2 y = 7 x - 10

The solutions are:

x = 4 , y = 9

The product of the digits:

x ∙ y = 4 ∙ 9 = 36

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2. Suppose the two digits are a and b. Then the value of the number is 10a+b.
So, the value after the digit swap is 10b+a
I assume "that number" refers to the number whose digits were swapped.

The amount of increase in the sum is (10b+a)-(10a+b)=9b-9a
Now we know that
9b-9a = 10a+b-4
3(a+b) = 10a+b-10

19a-8b = 4
7a-2b = 10
a=4 b=9

ab = 36

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3. Why (10b+a)-(10a+b) this step can you explain

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4. cmon. That is the difference between the original number and with its digits reversed. Better reread what I wrote.

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5. I don't understand the entire problem

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