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In Class IX, you began your exploration of the world of real numbers and encountered
irrational numbers. We continue our discussion on real numbers in this chapter. We
begin with two very important properties of positive integers in Sections 1.2 and 1.3,
namely the Euclid’s division algorithm and the Fundamental Theorem of Arithmetic.
Euclid’s division algorithm, as the name suggests, has to do with divisibility of
integers. Stated simply, it says any positive integer a can be divided by another positive
integer b in such a way that it leaves a remainder r that is smaller than b. Many of you
probably recognise this as the usual long division process. Although this result is quite
easy to state and understand, it has many applications related to the divisibility properties
of integers. We touch upon a few of them, and use it mainly to compute the HCF of
two positive integers.
The Fundamental Theorem of Arithmetic, on the other hand, has to do something
with multiplication of positive integers. You already know that every composite number
can be expressed as a product of primes in a unique way—this important fact is the
Fundamental Theorem of Arithmetic. Again, while it is a result that is easy to state and
understand, it has some very deep and significant applications in the field of mathematics.
- Euclid’s Division Lemma
Consider the following folk puzzle*.
A trader was moving along a road selling eggs. An idler who didn’t have
much work to do, started to get the trader into a wordy duel. This grew into a
fight, he pulled the basket with eggs and dashed it on the floor. The eggs broke.
The trader requested the Panchayat to ask the idler to pay for the broken eggs.
The Panchayat asked the trader how many eggs were broken. He gave the
If counted in pairs, one will remain;
If counted in threes, two will remain;
If counted in fours, three will remain;
If counted in fives, four will remain;
If counted in sixes, five will remain;
If counted in sevens, nothing will remain;
My basket cannot accomodate more than 150 eggs.
So, how many eggs were there? Let us try and solve the puzzle. Let the number
of eggs be a. Then working backwards, we see that a is less than or equal to 150:
If counted in sevens, nothing will remain, which translates to a = 7p + 0, for
some natural number p. If counted in sixes, a = 6 q +5, for some natural number q.
If counted in fives, four will remain. It translates to a = 5w + 4, for some natural
If counted in fours, three will remain. It translates to a = 4s + 3, for some natural
If counted in threes, two will remain. It translates to a = 3t + 2, for some natural
If counted in pairs, one will remain. It translates to a = 2u + 1, for some natural
That is, in each case, we have a and a positive integer b (in our example,
b takes values 7, 6, 5, 4, 3 and 2, respectively) which divides a and leaves a remainder
r (in our case, r is 0, 5, 4, 3, 2 and 1, respectively), that is smaller than b.
IMPORTANT FORMULA OF REAL NUMBER CLASS 10TH
Euclid’s Division Lemma Formula Of Real Number Class 10th
Euclid’s Division Lemma is defined as “for a given positive integer a and b, there exist unique integers q and r satisfying [a=bq+r,0≤r<b].
HCF (Highest common factor) Formula Of Real Number Class 10th
HCF of two positive integers can be find using the Euclid’s Division Lemma algorithm
We know that for any two integers a. b. we can write following expression
a=bq + r , 0≤r<b
If r=0 .then
HCF( a. b) =b
If r≠0 , then
HCF ( a. b) = HCF ( b.r)
Again expressing the integer b.r in Euclid’s Division Lemma, we get
b=pr + r1
HCF ( b,r)=HCF (r,r1)
Solved Example Of Real Number Class 10th
Example 1 : Use Euclid’s algorithm to find the HCF of 4052 and 12576.
Step 1 : Since 12576 > 4052, we apply the division lemma to 12576 and 4052, to get
12576 = 4052 × 3 + 420
Step 2 : Since the remainder 420 ≠ 0, we apply the division lemma to 4052 and 420, to
4052 = 420 × 9 + 272
Step 3 : We consider the new divisor 420 and the new remainder 272, and apply the
division lemma to get
420 = 272 × 1 + 148
We consider the new divisor 272 and the new remainder 148, and apply the division
lemma to get
272 = 148 × 1 + 124
We consider the new divisor 148 and the new remainder 124, and apply the division
lemma to get
148 = 124 × 1 + 24
We consider the new divisor 124 and the new remainder 24, and apply the division
lemma to get
124 = 24 × 5 + 4
We consider the new divisor 24 and the new remainder 4, and apply the division
lemma to get
24 = 4 × 6 + 0
The remainder has now become zero, so our procedure stops. Since the divisor at this
stage is 4, the HCF of 12576 and 4052 is 4.
Notice that 4 = HCF (24, 4) = HCF (124, 24) = HCF (148, 124) =
HCF (272, 148) = HCF (420, 272) = HCF (4052, 420) = HCF (12576, 4052).
Euclid’s division algorithm is not only useful for calculating the HCF of very
large numbers, but also because it is one of the earliest examples of an algorithm that
a computer had been programmed to carry out.
Example 2 : A sweetseller has 420 kaju barfis and 130 badam barfis. She wants to
stack them in such a way that each stack has the same number, and they take up the
least area of the tray. What is the number of that can be placed in each stack for this
Solution : This can be done by trial and error. But to do it systematically, we find
HCF (420, 130). Then this number will give the maximum number of barfis in each
stack and the number of stacks will then be the least. The area of the tray that is used
up will be the least.
Now, let us use Euclid’s algorithm to find their HCF. We have :
420 = 130 × 3 + 30
130 = 30 × 4 + 10
30 = 10 × 3 + 0
So, the HCF of 420 and 130 is 10.
Therefore, the sweetseller can make stacks of 10 for both kinds of barfi.
Some Question Of Real Number Class 10th
Q1. Use Euclid’s division algorithm to find the HCF of :
(i) 135 and 225 (ii) 196 and 38220 (iii) 867 and 255
Q2. Show that any positive odd integer is of the form 6q + 1, or 6q + 3, or 6q + 5, where q is
Q3. An army contingent of 616 members is to march behind an army band of 32 members in
a parade. The two groups are to march in the same number of columns. What is the
maximum number of columns in which they can march?
Q4. Use Euclid’s division lemma to show that the square of any positive integer is either of
the form 3m or 3m + 1 for some integer m.
[Hint : Let x be any positive integer then it is of the form 3q, 3q + 1 or 3q + 2. Now square
each of these and show that they can be rewritten in the form 3m or 3m + 1.]
Q5. Use Euclid’s division lemma to show that the cube of any positive integer is of the form
9m, 9m + 1 or 9m + 8.
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