(i) a^{m} * a^{n} = a^{m} + ^{n}
(ii) a^{m}/a^{n} = a^{m} - ^{n}
(iii) (a^{m})^{n} = a^{m} * ^{n}
(v) a^{-n} = 1/a^{n}
(vi) ^{n}$\sqrt{a}$^{m} = a^{m}/^{n}
(vii) (ab)^{m}= a^{m} ∙ b^{m}.
(viii) (a/b)^{m} = a^{m}/b^{m}
Divisibility rules:
Divisibility By 24 : A given number is divisible by 24, if it is divisible by both 3 and 8.
Divisibility By 40 : A given number is divisible by 40, if it is divisible by both 5 and 8.
Divisibility By 80 : A given number is divisible by 80, if it is divisible by both 5 and 16.
Important Notes :
If a number is divisible by "p" as well as "q", where "p" and "q" are co-primes, then the given number is divisible by "pq".
If p and q are not co-primes, then the given number need not be divisible by pq, even when it is divisible by both p and q.
Example :36 is divisible by both 4 and 6, but it is not divisible by (4x6) = 24, since 4 and 6 are not co-primes.
Ans .
yes
Sum of digits in 541326 = (5 + 4 + 1 + 3 + 2 + 6) = 21, which is divisible by 3. Hence, 541326 is divisible by 3
Ans .
no
Sum of digits in 5967013 = (5 + 9 + 6 + 7 + 0 + 1 + 3) = 31, which is not divisible by 3. Hence, 5967013 is not divisible by 3.
(a + b)^{2} = a^{2} + b^{2} + 2ab
(a - b)^{2} = a^{2} + b^{2} - 2ab
(a + b)^{2} - (a - b)^{2} = 4ab
(a + b)^{2} + (a + b)^{2} = 2 (a^{2} + b^{2})
(a^{2} - b^{2}) = (a + b) (a - b)
(a + b + c)^{2} = a^{2} + b^{2} + c^{2} + 2(ab + bc + ca)
(a^{3} + b^{3}) = (a + b)(a^{2} - ab + b^{2})
(a^{3} - b^{3}) = (a - b)(a^{2} + ab + b^{2})
(a^{3} + b^{3} + c^{3} - 3abc) = (a + b + c)(a^{2} + b^{2} + c^{2} - ab - bc - ac)
If a + b + c = 0, then a^{3} + b^{3} + c^{3} = 3abc
If we divide a given number by another number, then : Dividend = (Divisor x Quotient) + Remainder
(x^{n} - a^{n}) is divisible by (x - a) for all values of "n"
(x^{n} - a^{n}) is divisible by (x + a) for even values of "n"
(x^{n} + a^{n}) is divisible by (x + a) for odd values of "n"
Some Properties of Prime Numbers
The lowest prime number is 2. 2 is also the only even prime number. The lowest odd prime number is 3.
The remainder when a prime number p ≥ 5 is divided by 6 is 1 or 5. However, if a number on being divided by 6 gives a remainder of 1 or 5 the number need not be prime.
The remainder of the division of the square of a prime number p ≥ 5 divided by 24 is 1.
For prime numbers p > 3, p^{2} - 1 is divisible by 24.
If a and b are any two odd primes then a^{2} - b^{2} is composite. Also, a^{2} + b^{2} is composite.
The remainder of the division of the square of a prime number p ≥ 5 divided by 12 is 1
Shortcut to Test for Prime numbers
Step 1 : A number below 49 is prime if it is not divisible by 3 (Except numbers is even or ending with 0 / 5).
Step 2 : A number between 49 - 121 is prime if it is not divisible by 3 (Except even numbers or ending with 0 / 5 or the numbers 77, 91, 119).
Step 3 : A number between 121 - 169 is prime if it is not divisible by 3 (Except even numbers or numbers ending with 0 / 5 and 133, 143, 161).
Co prime or relatively prime: Two numbers are said to be relatively prime if they have only 1 as the common factor. E.g: (4,5); (8,9). If a number is divisible by p,q where both are co primes then it is also divisible by p*q;
Ans .
3572021
Let x - 1936248=1635773. Then, x = 1635773 + 1936248 = 3572021.
Ans .
5526
Let 8597 - x = 7429 - 4358. Then, x = (8597 + 4358) - 7429 = 12955 - 7429 = 5526.
Ans .
9
When we add the numbers together, at the units place we have 9 + 7 + 8 = 24. We carried the 2 to the tens place and got P + Q + R + 2, As we need 11 at the next digit we need to carry forward 1. So P + Q + R + 2 = 11 and we get Q as 9. P = R = 0
Ans .
57928256595
5793405 x 9999 = 5793405(10000-1) = 57934050000-5793405 = 57928256595
Ans .
524673750
839478 x 625 = 839478 x 5^{4} = 8394780000 / 16 = 524673750.
Ans .
986000
986 x 137 + 986 x 863 = 986 x (137 + 863) = 986 x 1000 = 986000
Ans .
98300
983 x 207 - 983 x 107 = 983 x (207 - 107) = 983 x 100 = 98300
Ans .
2576025
i) 1605 x 1605 = (1605)^{ 2} = (1600 + 5)^{ 2} = (1600)^{ 2} + (5)^{ 2} + 2 x 1600 x 5 = 2560000 + 25 + 16000 = 2576025.
Ans .
1954404
1398 x 1398 - (1398)^{ 2} = (1400 - 2)^{ 2} = (1400)^{ 2} + (2)^{ 2} - 2 x 1400 x 2 = 1960000 + 4 - 5600 = 1954404.
Ans .
180338
(a^{2} + b^{2}) = 1/2 [(a + b)^{2} + (a - b)^{2}] (313)^{2} + (287)^{2} = 1/2 [(313 + 287)^{2} + (313 - 287)^{2}] = 1/2 [(600)^{2} + (26)^{2}] = 1/2 (360000 + 676) = 180338.
Ans .
761200
= (896)^{2} - (204)^{2} = (896 + 204) (896 - 204) = 1100 x 692 = 761200.
Ans .
251001
(ii) Given exp = (387)^{2} + (114)^{2} + (2 x 387 x 114) = a^{2} + b^{2} + 2ab, where a = 387, b = 114 = (a+b)^{2} = (387 + 114)^{2} = (501)^{2} = 251001.
Ans .
169
Given exp = (81)^{2} + (68)^{2} – 2 x 81 x 68 = a^{2} + b^{2} – 2ab, Where a = 81, b = 68 = (a-b)^{2} = (81 – 68)^{2} = (13)^{2} = 169
Ans .
yes
Sum of digits in 541326 = (5 + 4 + 1 + 3 + 2 + 6) = 21, which is divisible by 3. Hence, 541326 is divisible by 3
Ans .
no
Sum of digits in 5967013 =(5+9 + 6 + 7 + 0+1 +3) = 31, which is not divisible by 3. Hence, 5967013 is not divisible by 3.
Ans .
2
Let the missing digit be x. Sum of digits = (1 + 9 + 7 + x + 5 + 4 + 6 + 2) = (34 + x). For (34 + x) to be divisible by 9, x must be replaced by 2 . Hence, the digit in place of * must be 2.
Ans .
no
The number formed by the last two digits in the given number is 94, which is not divisible by 4. Hence, 67920594 is not divisible by 4.
Ans .
yes
The number formed by the last two digits in the given number is 72, which is divisible by 4. Hence, 618703572 is divisible by 4.
Ans .
4 and 0
Since the given number is divisible by 5, so 0 or 5 must come in place of $. But, a number ending with 5 is never divisible by 8. So, 0 will replace $. Now, the number formed by the last three digits is 4*0, which becomes divisible by 8, if * is replaced by 4. Hence, digits in place of * and $ are 4 and 0 respectively.
Ans .
yes
(Sum of digits at odd places) - (Sum of digits at even places) = (8 + 7 + 3 + 4) - (1 + 2 + 8) = 11, which is divisible by 11. Hence, 4832718 is divisible by 11.
Ans .
yes
24 = 3 x 8, where 3 and 8 are co-primes. The sum of the digits in the given number is 36, which is divisible by 3. So, the given number is divisible by 3. The number formed by the last 3 digits of the given number is 744, which is divisible by 8. So, the given number is divisible by 8. Thus, the given number is divisible by both 3 and 8, where 3 and 8 are co-primes. So, it is divisible by 3 x 8, i.e., 24.
Ans .
2
On dividing 3000 by 19, we get 17 as remainder. Number to be added = (19 - 17) = 2.
Ans .
3
On dividing 3105 by 21, we get 18 as remainder. So Number to be added to 3105 = (21 - 18) = 3. Hence, required number = 3105 + 3 = 3108
Ans .
9
On dividing the given number by 342, let k be the quotient and 47 as remainder. Then, number – 342k + 47 = (19 x 18k + 19 x 2 + 9) = 19 (18k + 2) + 9. The given number when divided by 19, gives (18k + 2) as quotient and 9 as remainder.
Ans .
100011
Smallest number of 6 digits is 100000. On dividing 100000 by 111, we get 100 as remainder. So Number to be added = (111 - 100) = 11. Hence, required number = 100011
Ans .
179
Divisor = (Dividend - Remainder ) / Quotient = ( 15968-37 ) / 89 = 179
Ans .
6, 4, 2
Calculate the number of factors of a number
Step 1 : Calculate the factors of the number "X" less than or equal to its square root or if its not a perfect square than less than or equal to a number whose square is less but closest to it. Example: For 80 it will be 1 - 8 as 9
Step 2 : for factors above the square root limit, we just find the numbers with whom we can multiply the factors obtained from Step 1 and get the number "X". Example : For factors of 80, we find factors from 1 - 8, i.e. 1,2,4,5,8 then for factors above square root limit we get 80,40,20,16,10. As 1 * 80, 40 * 2, 20 * 4, 16 * 5 and 10 * 8.
Calculate the Sum of factors of a number:
Step 1: Get prime factors of a number say 240
240 = 2^{4} * 3^{1} * 5^{1}
Step 2: Sum of factors formula is
240 = (2^{0} + 2^{1} + 2^{2} + 2^{3} + 2^{4}) * (3^{0} + 3^{1}) * (5^{0} + 5^{1})
Step 3: 31*4*6 = 744
Calculate the Number of factors of a number:
Step 1: Get the prime factors of a number
240 = 2^{4} * 3^{1} * 5^{1}
Step 2: Number of factors of a number.
Number of factors = ( 4 + 1 ) * ( 1 + 1) * ( 1 + 1) = 5 * 2 * 2 = 20
Thus the powers of the numbers are increased by one and multiplied.
Calculate the sum and number of even factors of a number :
Step 1: Get the prime factors of a number
240 = 2^{4} * 3^{1} * 5^{1}
^{ }
Step 2: Sum of even factors
240 = 2^{4} * 3^{1} * 5^{1}
^{ }
Step 2: Sum of odd factors
30 = 2^{1} * 3^{1} * 5^{1}
^{ }
Step 2: Sum of factors formula is
30 = (2^{0} + 2^{1} ) * (3^{0} + 3^{1}) * (5^{0} + 5^{1})
Ans .
18
2450 = 50 * 49 = 2^{1} * 5^{2} * 7^{2} Sum and number of all factors: Sum of factors = (2^{0} + 2^{1}) (5^{0} + 5^{1} + 5^{2}) (7^{0} + 7^{1} + 7^{2}) Number of factors = 2 * 3 * 3 = 18
Ans .
9
Sum of all even factors: (2^{1}) (5^{0} + 5^{1} + 5^{2}) (7^{0} + 7^{1} + 7^{2}) Number of even factors = 1 * 3 * 3 = 9
Ans .
9
Sum of all odd factors: (2^{0}) (5^{0} + 5^{1} + 5^{2}) (7^{0} + 7^{1} + 7^{2}) Number of odd factors = 1 * 3 * 3 = 9
Ans .
12
Sum of factors divisible by 5: (2^{0} + 2^{1}) (5^{1} + 5^{2}) (7^{0} + 7^{1} + 7^{2}) Number of factors divisible by 5 = 2 * 2 * 3 = 12
Ans .
8
Sum of factors divisible by 35: (2^{0} + 2^{1}) (5^{1} + 5^{2}) (7^{1} + 7^{2}) Number of factors divisible by 5 = 2 * 2 * 2= 8
Ans .
4
Sum of factors divisible by 245: (2^{0} + 2^{1}) (5^{1} + 5^{2}) (7^{2}) Number of factors divisible by 5 = 2 * 2 * 1= 4
Ans .
54
Sum and number of all factors: Sum of factors = (2^{0} + 2^{1}+ 2^{2} + 2^{3} + 2^{4} + 2^{5}) (3^{0} + 3^{1} + 3^{2}) (5^{0} + 5^{1} + 5^{2}) Number of factors = 6 * 3 * 3 = 54
Ans .
30
Sum and number of all even factors: Sum of factors = (2^{1}+ 2^{2} + 2^{3} + 2^{4} + 2^{5}) (3^{0} + 3^{1} + 3^{2}) (5^{0} + 5^{1} + 5^{2}) Number of factors = 5 * 3 * 3 = 30
Ans .
9
Sum and number of all odd factors: Sum of factors = (2^{0}) (3^{0} + 3^{1} + 3^{2}) (5^{0} + 5^{1} + 5^{2}) Number of factors = 1 * 3 * 3 = 9
Ans .
18
Sum and number of factors divisible by 25: Sum of factors = (2^{0} + 2^{1}+ 2^{2} + 2^{3} + 2^{4} + 2^{5}) (3^{0} + 3^{1} + 3^{2}) (5^{2}) Number of factors = 6 * 3 * 1 = 18
Ans .
18
Sum and number of all factors divisible by 40: Sum of factors = (2^{3} + 2^{4} + 2^{5}) (3^{0} + 3^{1} + 3^{2}) (5^{1} + 5^{2}) Number of factors = 3 * 3 * 2 = 18
Ans .
10
Sum and number of factors divisible by 150: Sum of factors = (2^{1}+ 2^{2} + 2^{3} + 2^{4} + 2^{5}) ( 3^{1} + 3^{2}) (5^{2}) Number of factors = 5 * 2 * 1 = 10
Ans .
44
Sum and number of factors NOT divisible by 150 = Total factors - Factors divisible by 150 = 54 - 10 = 44
Ans .
12
Sum and number of all factors who are perfect squares: Sum of factors = (2^{0} + 2^{2} + 2^{4} ) (3^{0} + 3^{2}) (5^{0} + 5^{2}) Number of factors = 3 * 2 * 2 = 12
New Age Consultants have three consultants Gyani, Medha and Buddhi. The sum of the number of projects handled by Gyani and Buddhi individually is equal to the number of projects in which Medha is involved. All three consultants are involved together in 6 projects. Gyani works with Medha in 14 projects. Buddhi has 2 projects with Medha but without Gyani, and 3 projects with Gyani but without Medha. The total number of projects for New Age Consultants is one less than twice the number of projects in which more than one consultant is involved.
Q. What is the number of projects in which Gyani alone is involved?
Uniquely equal to zero.
Uniquely equal to 1.
Uniquely equal to 4
Cannot be determined uniquely.
Ans . D
Putting the value of M in either equation, we get G + B = 17
Hence neither of two can be uniquely determined.
Q. What is the number of projects in which Medha alone is involved?
Uniquely equal to zero.
Uniquely equal to 1.
Uniquely equal to 4
Cannot be determined uniquely.
Ans . B
G + B = M + 16 Also, M + B + G + 19 = (2 * 19) – 1
i.e. (G + B) = 18 – M Thus, M + 16 = 18 – M
i.e. M = 1
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