Solution:
Sqrt(551) > 22
All prime numbers less than 24 are : 2, 3, 5, 7, 11, 13, 17, 19, 23.
119 is divisible by 7; 187 is divisible by 11;
247 is divisible by 13 and 551 is divisible by 19.
So, none of the given numbers is prime.
2. The difference of the squares of two consecutive even integers is divisible by which of the following integers?
Solution:
Let the two consecutive even integers be 2n and (2n + 2).
Then the difference of squares of these two consecutive even integers = (2n + 2)^2 – (2n)^2
= (2n)^2 + 8n + 4 – (2n)^2
= 8n + 4
= 4(2n + 1)
=> which is divisible by 4
3. A number when divided by 6 leaves a remainder 3. When the square of the number is divided by 6, the remainder is:
Solution:
45 = 5 x 9, where 5 and 9 are co-primes.
Unit digit must be 0 or 5 and sum of digits must be divisible by 9.
Among given numbers, such number is 202860.
5. (x^n - a^n) is completely divisible by (x - a),when
Solution:
72 = 9 x8, where 9 and 8 are co-prime. The minimum value of x for which 73x for which 73x is divisible by 8 is, x = 6.
Sum of digits in 425736 = (4 + 2 + 5 + 7 + 3 + 6) = 27, which is divisible by 9. Required value of * is 6.
13. Which natural number is nearest to 8485, which is completely divisible by 75 ?
Solution:
Prime numbers less than 50 are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47 Their number is 15
16. A boy multiplied 987 by a certain number and obtained 559981 as his answer. If in the answer both 9 are wrong and the other digits are correct, then the correct answer would be:
Solution:
987 = 3 x 7 x 47 So, the required number must be divisible by each one of 3, 7, 47
553681 (Sum of digits = 28, not divisible by 3)
555181 (Sum of digits = 25, not divisible by 3)
555681 is divisible by 3, 7, 47.
17. On dividing 2272 as well as 875 by 3-digit number N, we get the same remainder. The sum of the digits of N is
Solution:
Clearly, (2272 - 875) = 1397, is exactly divisible by N. Now, 1397 = 11 x 127 The required 3-digit number is 127, the sum of whose digits is 10.
18. If the number 5 * 2 is divisible by 6, then * = ?
Solution:
Let the given number be 476 xy 0. Then (4 + 7 + 6 + x + y + 0) = (17 + x + y) must be divisible by 3. And, (0 + x + 7) - (y + 6 + 4) = (x - y -3) must be either 0 or 11.
x - y - 3 = 0 y = x - 3 (17 + x + y) = (17 + x + x - 3) = (2x + 14) x= 2 or x = 8. Therefore, x = 8 and y = 5.