Question

Five double-digit numbers We have five consecutive positive double-digit integers. If we swap places on the numbers in the largest number, the sum of the five numbers increases by 1 more than the mean of the original numbers. What is the lowest number?

Answer 1

• 33

Explanation:

Let the numbers are:

• x, x + 1, x + 2, x + 3 and x + 4

• Their sum is 5x + 10
• Their mean is (5x + 10)/2 = x + 2

If we swap the digits on the largest number, the sum increases by 1 more than x + 2

We are looking for the number x + 4 = ab such that:

• ba - ab = ab - 2 + 1
• ba = 2ab - 1
• 10b + a = 2(10a + b) - 1
• 10b + a = 20a + 2b - 1
• 8b = 19a - 1
• 19a = 8b + 1

By trial method we get the solution:

• a = 3, b = 7

Since x + 4 = 37, the lowest number is:

• x = 37 - 4 = 33

Lets verify:

• 33, 34, 35, 36, 37

The sum is:

• 5*33 + 10 = 175

The mean is:

• 35

Change the largest number tp 73 and find the sum again:

• 175 + (73 - 37) = 175 + 36 = 175 + 35 + 1

The sum has increased by 1 more than 35