<PRE><B>Question 1096585</B>
&lt;br&gt;There are two distinct possible approaches to solving this problem:&lt;br&gt;
(1) Use algebra to find what 2-digit numbers have the property that the number is 3 more than 4 times the sum of its digits, and find which one of those numbers satisfies the property that adding 18 to the number produces a number which has the same digits in reverse order; or&lt;br&gt;
(2) Finding the 2-digit numbers which have the property  that adding 18 to the number produces a number which has the same digits in reverse order, and finding the one for which the number is 3 more than 4 times the sum of its digits.&lt;br&gt;
If you know the right number trick, the second option will get you to the answer much faster than the first.  But both methods are valid; and taking a look at both of them can be useful exercises.  So...&lt;br&gt;
(1) using algebra....&lt;br&gt;
If the 2-digit number has tens digit x and units digit y, then the value of the 2-digit number is 10x+y.  We want that to be equal to 3 more than 4 times the sum of x and y:
{{{10x+y = 4(x+y)+3}}}
{{{10x+y = 4x+4y+3}}}
{{{6x = 3y+3}}}
{{{2x = y+1}}}
{{{y = 2x-1}}}&lt;br&gt;
We can easily list the combinations of x and y values that satisfy this equation and in which x and y are both single digits:
x=1 --&gt; y=1 --&gt; the 2-digit number is 11
x=2 --&gt; y=3 --&gt; the 2-digit number is 23
x=3 --&gt; y=5 --&gt; the 2-digit number is 35
x=4 --&gt; y=7 --&gt; the 2-digit number is 47
x=5 --&gt; y=9 --&gt; the 2-digit number is 59&lt;br&gt;
Of those 2-digit numbers, there is only one which, when 18 is added to it, produces a 2-digit number with the two digits reversed: 35+18 = 53.&lt;br&gt;
So the 2-digit number we are looking for is 35.&lt;br&gt;
(2) the alternative method...&lt;br&gt;
If you have a 2-digit number, and adding 18 to the number produces a 2-digit number with the digits reversed, then the number has a tens digit which is 2 less than the units digit.  The 2-digit numbers with this property are
13  24  35  46  57  68  79&lt;br&gt;
Checking these to find which of them has the property that the number is 3 more than 4 times the sum of its digits, we see that the only one satisfying that property is 35.&lt;br&gt;
So again by this method the number we are looking for is 35.

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