Number System - Study Mode

[#271] For an integer n, n! = n(n - 1) (n - 2) ..... 3.2.1 Then, 1! + 2! + 3! +.....+ 100!, when divided by 5 leaves remainder
Correct Answer

(D) 3

Explanation

Solution: Every number from 5 onwards is completely divisible by 5 ∴ ( 5 + 6 + 7 + ...... + 100 ) is completely divisible by 5 And, ( 1 + 2 + 3 + 4 ) $$eqalign{
& = left( {1 + 2 + 3 imes 2 imes 1 + 4 imes 3 imes 2 imes 1}
ight) cr
& = left( {1 + 2 + 6 + 24}
ight) cr
& = 33 cr} $$ Clearly, 33 when divided by 5 leave a remainder 3 Hence, ( 1 + 2 + 3 + 4 + 5 + ...... + 100 ) When divided by 5 leaves a remainder 3

[#272] A number is multiplied by 11 and 11 is added to the product. If the resulting number is divisible by 13, the smallest original number is = ?
Correct Answer

(A) 12

Explanation

Solution: Let the required number be x. Then, $$frac{11x + 11}{13}$$ xa0 = a whole number. So, (11x + 11) must be divisible by 13 By hit and trial, we get x = 12 Hence, the smallest original number is 12

[#273] The smallest prime number, that is the fifth term of an increasing arithmetic sequence in which all the four preceding terms are also prime, is -
Correct Answer

(B) 29

Explanation

Solution: The required arithmetic sequence of five prime numbers is 5, 11, 17, 23, 29 and therefore, the required 5 th term is 29

[#274] Among the following statements, the statement which is not correct is :
Correct Answer

(C) Every real number is a rational number.

Explanation

Solution: Every real number is a rational number is not a correct statements.

[#275] Which of the following numbers are completely divisible by 7 ? I. 195195 II. 181181 III. 120120 IV. 891891
Correct Answer

Only I and II

Explanation

Solution: I. We have (195 - 195) = 0 ∴ 195195 is divisible by 7 II. We have (181 - 181) = 0 ∴ 181181 is divisible by 7 III. We have (120 - 120) = 0 ∴ 120120 is divisible by 7 IV. We have (891 - 891) = 0 ∴ 891891 is divisible by 7 Hence, all are divisible by 7