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Question 1196802: If a number of two digits is divided by the sum of its digits, the quotient is 4 and the remainder is 6. If 9 is added to the number the sum has the same digits but inverted. What is the number?
Answer by math_tutor2020(3817)   (Show Source):
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Answer: 34
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Explanation:
a = tens digit
b = units digit
Each 'a' and b value is selected from the set {0,1,2,3,4,5,6,7,8,9} where repeats are possible.
The two digit number is of the form 10a+b
For example, if a = 7 is the tens digit and b = 5 is the units digit, then 10a+b = 10*7+5 = 75
Divide this over the sum of the digits a+b and we get a quotient of 4 and remainder 6
(10a+b)/(a+b) = 4 remainder 6
(10a+b)/(a+b) = 4 + 6/(a+b)
10a+b = 4(a+b) + 6
Let's solve for b
10a+b = 4(a+b) + 6
10a+b = 4a+4b + 6
b-4b = 4a+6-10a
-3b = -6a+6
b = (-6a+6)/(-3)
b = 2a-2
If 9 is added to the original number 10a+b, then we're told the digits swap.
original + 9 = swapped digits
(10a+b) + 9 = 10b+a
Now plug in b = 2a-2 and solve for 'a'
(10a+b) + 9 = 10b+a
(10a+2a-2) + 9 = 10(2a-2)+a
12a+7 = 20a-20+a
12a+7 = 21a-20
12a-21a = -20-7
-9a = -27
a = -27/(-9)
a = 3
Lastly, let's find b
b = 2a-2
b = 2*3-2
b = 6-2
b = 4
The original number is 10a+b = 10*3+4 = 34
Let's divide this over the sum of the digits a+b = 3+4 = 7
34/7 = 4 remainder 6
So far so good
Side note: you can think of it like you have 34 cookies and 7 friends to pass them out to.
Each friend gets 4 cookies so 7*4 = 28 cookies are taken so far.
That means there are 34-28 = 6 cookies leftover.
Now add 9 to the number
34+9 = 43
and the digit swap part is confirmed as well.
The answer is fully confirmed.
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