Page 79 - Ganit Kaushal

So, the problem can be solved by categorising it into two parts: leap years and non-leap years.

For a leap year, taking a starting day to be any day from Monday to Sunday gives 7 cases.
Again, for a non-leap year, taking the starting day to be any day from Monday to Sunday gives 7 cases. Therefore,
the total cases will be 7 + 7 = 14.
So, there are only 14 different calendars for all the years to come or those that have passed.

This gave the answer to all his questions.
Ex 1. The time now is 10:01. How many minutes will there be until the clock shows the next palindromic
time? What about the one after that? (NCERT)
Sol. There are 70 minutes until the clock shows the next palindromic time, and that is 11:11. The one after
that will again be after 70 minutes and that will be 12:21. Amazing! Isn’t it?
Ex 2. What is the sum of the smallest and largest 5-digit palindromes? What is the difference? (NCERT)
Sol. The largest 5-digit palindrome is 99999 and the smallest 5-digit palindrome is 10001.
Their sum = 99999 + 10001 = 1,10,000; Difference = 99999 – 10001 = 89998.

Exercise 3.4

1. How many rounds of the Kaprekar procedure does the number 5683 take to reach the Kaprekar constant?
(NCERT)
2. Pratibha uses the digits ‘4’, ‘7’, ‘3’ and ‘2’, and makes the smallest and largest 4-digit numbers with
them: 2347 and 7432. The difference between these two numbers is 7432 – 2347 = 5085. The sum of
these two numbers is 9779. (NCERT)
Choose 4 digits to make:
(a) the difference between the largest and smallest numbers greater than 5085.
(b) the difference between the largest and smallest numbers less than 5085.
(c) the sum of the largest and smallest numbers greater than 9779.

(d) the sum of the largest and smallest numbers less than 9779.
3. For three-digit numbers with not all digits identical, what is the number that starts repeating after a
few rounds of the Kaprekar procedure?
4. Complete the adjoining table by writing a number according to the Indian system of numeration. Also,
write the number in words. The number has to tell the following about itself. TTh Th H T U
(a) I am a 5-digit palindrome.
(b) I am an odd number. Write the number in words:
(c) My ‘T’ digit is double of my ‘U’ digit.
(d) My ‘H’ digit is double of my ‘T’ digit. (NCERT)
5. Write all palindromic calendar dates between the years 2010 and 2020.
6. Which 5-digit palindrome has the sum of its digits equal to 24 and the product of its digits equal to
zero?
7. The number 56665 is a palindrome. The next palindrome is 56765. Write this sequence 56665,
56765…… of palindromes up to eight terms. What is the 8th term of this sequence?
8. Which two palindromes, less than 1000, are at a difference of 2 only?

9. Which three-digit palindrome has the sum of its digits equal to 7 and product equal to 12?

Number Play 77