Answer to Question #329492 in Discrete Mathematics for Shang

Question

Solve each of the following problems. Follow Polya's four stages of problem solving in your solution.

3. A palindromic number or numeral palindrome is a "symmetrical" number like 16461, which remains the same when its digits are reversed. The term palindromic is derived from palindrome, which refers to a word like rotor that remains unchanged under reversal of its letters. How many palindromes are there between 0 and 1000?

4. Matt is half as old as James will be when Matt is twice as old as James is now. In five years, the sum of Matt's and James' ages will be 100. How old are Matt and James now?

5. Find the digit 120 places to the right of the decimal point in the decimal representation of 9/13

Answer 1

3) All one-digit numbers (from 0 to 9, their amount is 10) are palindromes.

Two-digit palindrome can be constructed by duplicating some digit except 0, so amount of them is 9.

Three-digit numbers can be constructed by inserting any digit between digits of some two-digit palindrome, so for each two-digit palindrome there are 10 corresponding three-digit palindromes, so amount of three-digit palindromes is 9 * 10 = 90.

Total amount of palindromes between 0 and 1000 is 10 + 9 + 90 = 109.

4) Let m - age of Matt, j - age of James.

Matt will be twice as old as James is now in (2j - m) years

At this point in time James will be j + (2j - m) = 3j - m years old

Matt is half as old as James will be when Matt is twice as old as James is now =>

=> m = (3j - m)) / 2 => 2m = 3j - m => 3m = 3j => m = j, so Matt and James are the same age.

Second condition: In five years, the sum of Matt's and James' ages will be 100 => m + 5 + j + 5 = 100 => m + j = 90 => 2m = 90 => m = 45, j = 45.

Matt and James are both 45 years old.

Let's try to divide 9 by 13:

$\underline{9~~~}|\underline{~13~~~~~~~~~~~~}\\ 90~|~0.692307\\ \underline{78~~}\\ 120\\ \underline{117~~}\\ ~~~~30\\ ~~~~\underline{26~~}\\ ~~~~~~40\\ ~~~~~~\underline{39~~~~}\\ ~~~~~~~~100\\ ~~~~~~~~~~\underline{91}\\ ~~~~~~~~~~~~9$

We can see that after getting 6 digits after comma we again got 9, so in further dividing we will get the same digits in cycle with period of 6: 0.692307692307... = 0.(692307).

120'th digit to the right will be the same as 6'th because 120 contains whole 20 6-digit-sized cycles. This digit is 7.