Basic concepts, definitions and identities
Number System :
Test of divisibility:
1. A number is divisible by ‘2’ if it ends in zero or in a digit which is a multiple of ‘2’i.e. 2,4, 6, 8.
2. A number is divisible by ‘3’, if the sum of the digits is divisible by ‘3’.
3. A number is divisible by ‘4’ if the number formed by the last two digits, i.e. tens and units are divisible by 4.
4. A number is divisible by ‘5’ if it ends in zero or 5
5. A number is divisible by ‘6’ if it divisible by ‘2’ as well as by ‘3’.
6. A number is divisible by ‘8’ if the number formed by the last three digits, i.e, hundreds tens and units is divisible by ‘8’.
7. A number is divisible by ‘9’ if the sum of its digit is divisible by ‘9’
8. A number is divisible by ‘10’ if it ends in zero.
9. A number is divisible by ‘11’ if the difference between the sums of the digits in the even and odd places is zero or a multiple of ‘11’.
LCM:
LCM of a given set of numbers is the least number which is exactly divisible by every number of the given set.
HCF:
HCF of a given set of numbers is the highest number which divides exactly every number of the given set.
Examples to Follow:
1. The square of an odd number is always odd.
2. A number is said to be a prime number if it is divisible only by itself and unity. Ex. 1, 2, 3, 5,7,11,13 etc.
3. The sum of two odd number is always even.
4. The difference of two odd numbers is always even.
5. The sum or difference of two even numbers is always even.
6. The product of two odd numbers is always odd.
7. The product of two even numbers is always even.
Problems:
1. If a number when divided by 296 gives a remainder 75, find the remainder when 37 divides the same number.
Method:
Let the number be ‘x’, say
∴x = 296k + 75, where ‘k’ is quotient when ‘x’ is divided by ‘296’
= 37 × 8k + 37 × 2 + 1
= 37(8k + 2) + 1
Hence, the remainder is ‘1’ when the number ‘x’ is divided by 37.
2. If 232+1 is divisible by 641, find another number which is also divisible by ‘641’.
= (232 +1)(264-232 +1)
From the above equation, we find that 296+1 is also exactly divisible by 641, since it is already given that 232+1 is exactly divisible by ‘641’.
3. If m and n are two whole numbers and if mn = 25. Find nm, given that n ≠1
Method:
m^n = 25 = 5^2
∴m = 5, n = 2
∴n^m = 2^5 = 32
4. Find the number of prime factors of 610 × 717 × 5527 610 × 717 × 5527 = 210×310×717×527×1127
Ans:
∴The number of prime factors = the sum of all the indices viz., 10 + 10 + 17 + 27 + 27 = 91
5. A number when successively divided by 9, 11 and 13 leaves remainders 8, 9 and 8 respectively. Method:
The least number that satisfies the condition= 8 + (9×9) + (8×9×11)
= 8 + 81 + 792 = 881
6. A number when divided by 19, gives the quotient 19 and remainder 9. Find the number.
Ans:
Let the number be ‘x’ say.
x = 19 × 19 + 9
= 361 + 9 = 370
7. Four prime numbers are given in ascending order of their magnitudes, the product of the first three is 385 and that of the last three is 1001. Find the largest of the given prime numbers.
Ans:
The product of the first three prime numbers = 385
The product of the last three prime numbers = 1001
In the above products, the second and the third prime numbers occur in common.
∴ The product of the second and third prime numbers = HCF of the given products.
HCF of 385 and 1001 = 77
∴Largest of the given primes = 1001/77
= 13
Square root, Cube root, Surds and Indices
Characteristics of square numbers
1. A square cannot end with an odd number of zeros
2. A square cannot end with an odd number 2, 3, 7 or 8
3. The square of an odd number is odd
4. The square of an even number is even.
5. Every square number is a multiple of 3 or exceeds a multiple of 3 by unity. Ex. 4 × 4 = 16 = 5 × 3 + 1 5 × 5 = 25 = 8 × 3 + 1 7 × 7 = 49 = 16 × 3 + 1
6. Every square number is a multiple of 4 or exceeds a multiple of 4 by unity. Ex. 5 × 5 = 25 = 6 × 4 + 1 7 × 7 = 49 = 12 × 4 + 1
7. If a square numbers ends in ‘9’, the preceding digit is even. Ex. 7 × 7 = 49 ‘4’ is the preceding even numbers 27 × 27 = 729 ‘2’ is the preceding even numbers.
Characteristics of square roots of numbers
1. If a square number ends in ‘9’, its square root is a number ending in’3’ or ‘7’.
2. If a square number ends in ‘1’, its square root is a number ending in’1’ or ‘9’.
3. If a square number ends in ‘5’, its square root is a number ending in’5’
4. If a square number ends in ‘4’, its square root is a number ending in’2’ or ‘8’.
5. If a square number ends in ‘6’, its square root is a number ending in’4’ or ‘6’.
6. If a square number ends in ‘0’, its square root is a number ending in ‘0’.
Number System :
Test of divisibility:
1. A number is divisible by ‘2’ if it ends in zero or in a digit which is a multiple of ‘2’i.e. 2,4, 6, 8.
2. A number is divisible by ‘3’, if the sum of the digits is divisible by ‘3’.
3. A number is divisible by ‘4’ if the number formed by the last two digits, i.e. tens and units are divisible by 4.
4. A number is divisible by ‘5’ if it ends in zero or 5
5. A number is divisible by ‘6’ if it divisible by ‘2’ as well as by ‘3’.
6. A number is divisible by ‘8’ if the number formed by the last three digits, i.e, hundreds tens and units is divisible by ‘8’.
7. A number is divisible by ‘9’ if the sum of its digit is divisible by ‘9’
8. A number is divisible by ‘10’ if it ends in zero.
9. A number is divisible by ‘11’ if the difference between the sums of the digits in the even and odd places is zero or a multiple of ‘11’.
LCM:
LCM of a given set of numbers is the least number which is exactly divisible by every number of the given set.
HCF:
HCF of a given set of numbers is the highest number which divides exactly every number of the given set.
Examples to Follow:
1. The square of an odd number is always odd.
2. A number is said to be a prime number if it is divisible only by itself and unity. Ex. 1, 2, 3, 5,7,11,13 etc.
3. The sum of two odd number is always even.
4. The difference of two odd numbers is always even.
5. The sum or difference of two even numbers is always even.
6. The product of two odd numbers is always odd.
7. The product of two even numbers is always even.
Problems:
1. If a number when divided by 296 gives a remainder 75, find the remainder when 37 divides the same number.
Method:
Let the number be ‘x’, say
∴x = 296k + 75, where ‘k’ is quotient when ‘x’ is divided by ‘296’
= 37 × 8k + 37 × 2 + 1
= 37(8k + 2) + 1
Hence, the remainder is ‘1’ when the number ‘x’ is divided by 37.
2. If 232+1 is divisible by 641, find another number which is also divisible by ‘641’.
Method:
Consider 296+1 = (232)3 + 13 = (232 +1)(264-232 +1)
From the above equation, we find that 296+1 is also exactly divisible by 641, since it is already given that 232+1 is exactly divisible by ‘641’.
3. If m and n are two whole numbers and if mn = 25. Find nm, given that n ≠1
Method:
m^n = 25 = 5^2
∴m = 5, n = 2
∴n^m = 2^5 = 32
4. Find the number of prime factors of 610 × 717 × 5527 610 × 717 × 5527 = 210×310×717×527×1127
Ans:
∴The number of prime factors = the sum of all the indices viz., 10 + 10 + 17 + 27 + 27 = 91
5. A number when successively divided by 9, 11 and 13 leaves remainders 8, 9 and 8 respectively. Method:
The least number that satisfies the condition= 8 + (9×9) + (8×9×11)
= 8 + 81 + 792 = 881
6. A number when divided by 19, gives the quotient 19 and remainder 9. Find the number.
Ans:
Let the number be ‘x’ say.
x = 19 × 19 + 9
= 361 + 9 = 370
7. Four prime numbers are given in ascending order of their magnitudes, the product of the first three is 385 and that of the last three is 1001. Find the largest of the given prime numbers.
Ans:
The product of the first three prime numbers = 385
The product of the last three prime numbers = 1001
In the above products, the second and the third prime numbers occur in common.
∴ The product of the second and third prime numbers = HCF of the given products.
HCF of 385 and 1001 = 77
∴Largest of the given primes = 1001/77
= 13
Square root, Cube root, Surds and Indices
Characteristics of square numbers
1. A square cannot end with an odd number of zeros
2. A square cannot end with an odd number 2, 3, 7 or 8
3. The square of an odd number is odd
4. The square of an even number is even.
5. Every square number is a multiple of 3 or exceeds a multiple of 3 by unity. Ex. 4 × 4 = 16 = 5 × 3 + 1 5 × 5 = 25 = 8 × 3 + 1 7 × 7 = 49 = 16 × 3 + 1
6. Every square number is a multiple of 4 or exceeds a multiple of 4 by unity. Ex. 5 × 5 = 25 = 6 × 4 + 1 7 × 7 = 49 = 12 × 4 + 1
7. If a square numbers ends in ‘9’, the preceding digit is even. Ex. 7 × 7 = 49 ‘4’ is the preceding even numbers 27 × 27 = 729 ‘2’ is the preceding even numbers.
Characteristics of square roots of numbers
1. If a square number ends in ‘9’, its square root is a number ending in’3’ or ‘7’.
2. If a square number ends in ‘1’, its square root is a number ending in’1’ or ‘9’.
3. If a square number ends in ‘5’, its square root is a number ending in’5’
4. If a square number ends in ‘4’, its square root is a number ending in’2’ or ‘8’.
5. If a square number ends in ‘6’, its square root is a number ending in’4’ or ‘6’.
6. If a square number ends in ‘0’, its square root is a number ending in ‘0’.
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