Objective:

This topic gives an overview of; 

• Numbers in General Form
• Games with Numbers
• Letters for Digits

Introduction

You have studied various types of numbers such as natural numbers, whole numbers, integers and rational numbers. You have also studied a number of interesting properties about them. In previous classes, we explored finding factors and multiples and the relationships among them.

In this chapter, we will explore numbers in more detail. These ideas help in justifying tests of divisibility.

Numbers in General Form

Let us take the number 52 and write it as

 52 = 50 + 2 = 10 × 5 + 2

Similarly, the number 37 can be written as

37 = 10 × 3 + 7

In general, any two digit number ab made of digits a and b can be written as

ab = 10 × a + b = 10a + b
 ba = 10 × b + a = 10b + a

Let us now take number 351. This is a three digit number. It can also be written as

351 = 300 + 50 + 1 = 100 × 3 + 10 × 5 + 1 × 1
497 = 100 × 4 + 10 × 9 + 1 × 7

Similarly, In general, a 3-digit number abc made up of digits a, b and c is written as

abc = 100 × a + 10 × b + 1 × c
 = 100a + 10b + c

In the same way,                                                   

cab = 100c + 10a + b
bca = 100b + 10c + a and so on.

Games with Numbers

 Reversing the digits – two digit number

Minakshi asks Sundaram to think of a 2-digit number, and then to do whatever she asks him to do, to that number. Their conversation is shown in the following figure. 

It so happens that Sundaram chose the number 49. So, he got the reversed number 94; then he added these two numbers and got 49 + 94 = 143. Finally he divided this number by 11 and got 143 ÷ 11 = 13, with no remainder. This is just what Minakshi had predicted.

Now, let us see if we can explain Minakshi’s “trick”.

Suppose Sundaram chooses the number ab, which is a short form for the 2-digit number 10a + b. On reversing the digits, he gets the number ba = 10b + a. When he adds the two numbers he gets:

(10a + b) + (10b + a) = 11a + 11b = 11 (a + b).

So, the sum is always a multiple of 11, just as Minakshi had claimed. Observe here that if we divide the sum by y 11, the quotient is a + b, which is exactly the sum of the digits of chosen number ab. Let us see how Sundaram explains Minakshi’s second “trick”.

Suppose he chooses the 2-digit number ab = 10a + b. After reversing the digits, he gets the number ba = 10b + a. Now Minakshi tells him to do a subtraction, the smaller number from the larger one.

• If the tens digit is larger than the ones digit (that is, a > b), he does:

(10a + b) – (10b + a) = 10a + b – 10b – a = 9a – 9b = 9(a – b).

• If the ones digit is larger than the tens digit (that is, b > a), he does:

(10b + a) – (10a + b) = 9(b – a).

•  And, of course, if a = b, he gets 0. 

In each case, the resulting number is divisible by 9. So, the remainder is 0. Observe here that if we divide the resulting number (obtained by subtraction), the quotient is a – b or b – a according as a > b or a < b. You may check the same by taking any other two digit numbers.

Reversing the digits – three digit number

Think of a 3-digit number, Now make a new number by putting the digits in reverse order, and subtract the smaller number from the larger one. Divide your answer by 99.There will be no remainder!!!

In fact, Minakshi chose the 3-digit number 349. So she got:

• Reversed number: 943;
• Difference: 943 – 349 = 594;
• Division: 594 ÷ 99 = 6, with no remainder.

Let us see how this trick works.

Let the 3-digit number chosen by Minakshi be abc = 100a + 10b + c. After reversing the order of the digits, she gets the number cba = 100c + 10b + a. On subtraction:

• If a > c, then the difference between the numbers is

(100a + 10b + c) – (100c + 10b + a) = 100a + 10b + c – 100c – 10b – a
= 99a – 99c = 99(a – c).

• If c > a, then the difference between the numbers is

(100c + 10b + a) – (100a + 10b + c) = 99c – 99a = 99(c – a).

• And, of course, if a = c, the difference is 0.

In each case, the resulting number is divisible by 99. So the remainder is 0. Observe that quotient is a – c or c – a. You may check the same by taking other 3-digit numbers.

Forming three-digit numbers with given three-digits

Think of any 3-digit number. Now use this number to form two more 3-digit numbers, like this:if the number you chose is abc, then

• ‘the first number is cab (i.e., with the ones digit shifted to the “left end” of the number);
• the other number is bca (i.e., with the hundreds digit shifted to the “right end” of the number).
• Now add them up. Divide the resulting number by 37.

In fact, Sundaram had thought of the 3-digit number 237. After doing what Minakshi had asked, he got the numbers 723 and 372. So he did: 

Then he divided the resulting number 1332 by 37: 1332 ÷ 37 = 36, with no remainder

Letters for Digits

Here we have puzzles in which letters take the place of digits in an arithmetic ‘sum’, and the problem is to find out which letter represents which digit; so it is like cracking a code. Here we stick to problems of addition and multiplication

Here are two rules we follow while doing such puzzles.

1. Each letter in the puzzle must stand for just one digit. Each digit must be represented by just one letter.
2. The first digit of a number cannot be zero. Thus, we write the number “sixty three” as 63, and not as 063, or 0063.

 A rule that we would like to follow is that the puzzle must have just one answer

Example: Find the digits A and B

 

Solution: This also has two letters A and B whose values are to be found.  Since the ones digit of 3 × A is A, it must be that A = 0 or A = 5. 

Now look at B. If B = 1, then BA × B3 would at most be equal to 19 × 19; that is, it would at most be equal to 361. But the product here is 57A, which is more than 500. So we cannot have B = 1.

If B = 3, then BA × B3 would be more than 30 × 30; that is, more than 900. But 57A is less than 600. So, B can not be equal to 3. Putting these two facts together, we see that B = 2 only. So the multiplication is either 20 × 23, or 25 × 23. The first possibility fails, since 20 × 23 = 460. But, the second one works out correctly, since 25 × 23 = 575. So the answer is A = 5, B = 2.

Summary

• Numbers can be written in general form. Thus, a two digit number ab will be written as ab = 10a + b.
• The sum of a 2-digit number and the number obtained by interchanging its digits is always divisible by 11.
• The difference between a 2-digit number and the number obtained by interchanging its digits is always divisible by 9.
• The general form of a 3-digit number is 100a +10b + c.
• The difference between a 3-digit number and a number obtained by reversing its digits is always divisible by 99.
• The general form of numbers are helpful in solving puzzles or number games.