When a certain

Question

When a certain three digit number is divided by the number with digits reversed, the quotient is 2 and the remainder is 25. The tens digit is one less than twice the sum of the hundreds digit and units digit. If the units digit is subtracted from the tens digit, the result is twice the hundreds digit. Find the number

Answer 1

The first trick is to write expressions for the two 3-digit numbers.

The digits are a, b and c.

(Eqn 1) abc = 100a + 10b +c

(Eqn 2) cba = 100c +10b + a

If you divide (1) by (2), you get 2 with a remainder of 25.

We can't do much with that yet, so let's see what else we know.

 

"The tens digit is one less than twice the sum of the hundreds digit and units digit."

"The tens digit ..."  --> b

"... is ..."  --> =

"one less than ..." --> We're going to subtract 1 from something.

"... twice ..."  --> 2 times something.

... the sum of the hundreds digit and units digit."  --> (a+c)

 

Put these pieces together:

(Eqn 3) b = 2(a+c)-1

 

"If the units digit (c) is subtracted from the tens digit (b), the result is (=) twice the hundreds digit (2a)."

(Eqn 4) b-c = 2a

 

Solve (4) for b and substitute that into (3)

(4) b = 2a + c

(3) 2a+c = 2(a+c)-1

2a+c = 2a+2c-1

-c = -1

c=1    The last digit is 1.

 

Let's use this to simplify (3)

(3)   b = 2(a+1)-1

b = 2a+1

 

Now we have an expression for b in terms of a.  We can use that and the known value for c to simplify the first two equations.

 

(1) 100a + 10b +c

100a+10(2a+1)+1

100a+20a+10+1

120a+11

 

(2) 100c +10b + a

100(1)+10(2a+1)+a
100+20a+10+a
21a+110

 

This is the really tricky spot!

Imagine you're doing long division of (Eqn 1) / (Eqn 2).

In the first step, you get 2.  Then you subtract 2 times the divisor from the numerator and that leaves a remainder of 25.

That's (Eqn 1) - 2(Eqn 2) = 25.

(120a+11)-2(21a+110) = 25

120a+11-42a-220 = 25

78a = 25+220-11

78a = 234

a = 3

 

Now we can use one of the simple equations to find b. I'll use the simplified version of Eqn 3:

b = 2a+1

b = (2)(3) + 1

b = 7

 

The number is 371.

 

Do the long division of 371 / 173 to check the result.  You get 2 with a remainder of 25.