☰
Real Numbers
Use Euclid’s division lemma to show that the cube of any positive integer is of the form 9m, 9m + 1 or 9m + 8.
Let a and b be two positive integergs such that a is greater than b; then :
a = bq + r;
where q and r are positive integers 0 ≤ r < b.
Taking b = 3, we get
a = 3q + r ; where 0 ≤ r < 3.
⇒ Different values of integer a are 3q, 3q + 1 or 3q + 2.
Cube of 3q = (3q)3
= 27q3 = 9(3q3) = 9m ;
where m is some integer.
Cube of 3q + 1 = (3q + 1)3
= (3q + 3(3q)2 × 1 + 3(3q) × 12+ l3
[∵ (a + b)3 = a3 + 3a2b + 3ab2 + 1]
= 27q3 + 27q2 + 9q + 1
= 9(3q3 + 3q2 + q) + 1
= 9m + 1; where m is some integer.
Cube of 3q + 2 = (3q + 2)3
= (3q)3 + 3(3q)2 × 2 + 3 × 3q × 22 + 23
= 27q3 + 54q2 + 36q + 8
= 9(3q3 + 6q2 + 4q) + 8 = 9m + 8; where m is some integer.
∴ Cube of any positive integer is of the form 9m or 9m + 1 or 9m + 8.
Sponsor Area
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.
Use Euclid’s division lemma to show that the cube of any positive integer is of the form 9m, 9m + 1 or 9m + 8.
Express each number as a product of its prime factors: (i) 140
Express each number as a product of its prime factors: (ii) 156
Express each number as a product of its prime factors: (ii) 156 (iii) 3825 (iv) 5005 (v) 7429
Express each number as a product of its prime factors: (iv) 5005
Express each number as a product of its prime factors: (v) 7429
Find the LCM and HCF of the following pairs of integers and verify that LCM × HCF = product of the two numbers. (i) 26 and 91
Find the LCM and HCF of the following pairs of integers and verify that LCM × HCF = product of the two numbers. (ii) 510 and 92
Find the LCM and HCF of the following pairs of integers and verify that LCM × HCF = product of the two numbers. (iii) 336 and 54
Sponsor Area
Sponsor Area