Number System


  1. A number is greater than the square of 44 but smaller than the square of 45. If one part of the number is the square of 6 and the number is a multiple of 5, then find the number. ?









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    Let the number be x.
    442 < x < 452
    ⇒ 1936 < x < 2025 ...(i)

    From equation (i), the required number will be any number between 1936 and 2025. Since one part of the number is the square of 6 means one factor is 36.

    ∴ L, C, M of 36 and 5 = 180
    ∴ Number will be multiple of 180 i,e.,
    180 x 11 = 1980 the only value which satisfies the equation (i).

    Correct Option: C

    Let the number be x.
    442 < x < 452
    ⇒ 1936 < x < 2025 ...(i)

    From equation (i), the required number will be any number between 1936 and 2025. Since one part of the number is the square of 6 means one factor is 36.

    ∴ L, C, M of 36 and 5 = 180
    ∴ Number will be multiple of 180 i,e.,
    180 x 11 = 1980 the only value which satisfies the equation (i).


  1. When a certain number is multiplied by 13, the product consists entirely of fives. The smallest such number is ?









  1. View Hint View Answer Discuss in Forum

    By trial, we find that the smallest number consisting entirely of fives and exactly divisible by 13 is 555555. On dividing 555555 by 13, we get 42735 as quotient.
    ∴ Req. smallest number =42735.

    Correct Option: C

    By trial, we find that the smallest number consisting entirely of fives and exactly divisible by 13 is 555555. On dividing 555555 by 13, we get 42735 as quotient.
    ∴ Req. smallest number =42735.



  1. A number consist of two digit whose sum is 10 If the digit of the number are reversed then the number decreased by 36. Which of the following is correct.

    I. The number is divisible by a composite number.
    II. The number is a multiple of a prime number.









  1. View Hint View Answer Discuss in Forum

    let the two-digit number be 10p + q
    Now according to the question
    p + q = 10
    and (p +10q) + 36 = (q + 10p)
    ⇒ -9q + 9p = 36
    ⇒ -q + p = 4 ....(ii)

    Correct Option: B

    let the two-digit number be 10p + q
    Now according to the question
    p + q = 10
    and (p +10q) + 36 = (q + 10p)
    ⇒ -9q + 9p = 36
    ⇒ -q + p = 4 ....(ii)
    on adding eq (i) and (ii) we get
    2p = 14 ⇒ p =7
    ∴ p = 7 and q = 4

    ∴ Required number is the 73
    So, the number is a multiple of prime number.


  1. Every prime number of the form 3k + 1 can be represented in the form 6m + 1 (where k, m are integers), When









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    The only one even prime number is 2 and this is not in the form of 3k + 1.
    Thus any prime number of the form 3k + 1 is an odd number.

    Correct Option: B

    The only one even prime number is 2 and this is not in the form of 3k + 1.
    Thus any prime number of the form 3k + 1 is an odd number.
    This means 3k must be even number which implies that k must be even number also.
    Let k = 2m.
    Then a prime of the form 3k + 1 is of the form 3(2m) + 1 = 6m+1.
    Every prime number of form 3k + 1 can be represented in the form 6m + 1 only, when k is even.



  1. A number consist of two digit whose sum is 10. If the digits of the number are reversed, then the number decreased by 36. Which of the following is/are correct?
    I. The number is divisible by a composite number.
    II. The number is multiple of a prime number.









  1. View Hint View Answer Discuss in Forum

    Let the two digit number be 10x + y.
    Now, according to the question,
    x + y = 10............(i)
    if digits reversed , number get decreased by 36,
    and ( x +10y) = ( 10x + y ) - 36
    Solve the equation.

    Correct Option: B

    Let the two digit number be 10x + y.
    Now, according to the question,
    x + y = 10............(i)
    if digits reversed , number get decreased by 36,
    and ( x +10y) = ( 10x + y ) - 36
    ⇒ - 9y + 9x = 36
    ⇒ x - y = 4 ..................... (ii)
    On adding Eqs. (i) and (ii), we get
    2x = 14
    ⇒ x = 7
    ∴ x = 7 and y = 3
    ∴ Required number is 73.
    So,the number is a multiple of a prime number.