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Numbers Exercise 3

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  1. Question 1 of 27
    1. Question

    If (74)² is subtracted from the square of a number, the answer so obtained is 5340. What is the number?

    Hint

    Let the number be x.

    Then, x² – (74)² = 5340

    ⇒ x² = 5340 + 5476 = 10816

    ⇒ x = number-system-q-49203.png

  2. Question 2 of 27
    2. Question

    If (74)² is subtracted from the square of a number, the answer so obtained is 3740. What is the number?

    Hint

    Let the number be = x

    According to the question

    x² – (74)² = 3740

    ⇒ x² = 3740 + 5476 = 9216

    ∴ x = number-system-q-49194.png = 96

  3. Question 3 of 27
    3. Question

    If (46)² is subtracted from the square of a number, the answer so obtained is 485. What is the number?

    Hint

    Let the number be x

    ∴ x² – (46)² = 485

    ⇒ x² = 485 + (46)² = 2601

    ∴ x = number-system-q-49185.png = 51

  4. Question 4 of 27
    4. Question

    If (57)² is added to the square of a number, the answer so obtained is 8010. What is the number?

    Hint

    Let the number be = x

    According to the question,

    x² + 57² = 8010

    ⇒ x² + 3249 = 8010

    ⇒ x² = 8010 – 3249 = 4761

    ⇒ x = number-system-q-49175.png = 69

  5. Question 5 of 27
    5. Question

    If (9)³ is subtracted from the square of a number, the answer so obtained is 567. What is the number?

    Hint

    Let the required number be x

    ∴ x² – (9)³ = 567

    x² = 567 + 729 = 1296

    ∴ x = number-system-q-49167.png = 36

  6. Question 6 of 27
    6. Question

    If (78)² is subtracted from the square of the number, the answer so obtained is 6,460. What is the number?

    Hint

    Let the number be x.

    According to the question,

    x² – 78² = 6460

    ⇒ x² = 6460 + 6084

    ⇒ x² = 12544

  7. Question 7 of 27
    7. Question

    What is the least number to be added to 4400 to make it a perfect square?

    Hint

    number-system-q-49157.png = 66.33

    ∴ Required number = 67² – 4400

    = 4489 – 4400 = 89

    ⇒ x = number-system-q-49148.png = 112

  8. Question 8 of 27
    8. Question

    Which least number shall be added to 8115 to make it a perfect square?

    Hint

    number-system-q-49142.png

    ∴ required number = 91 × 91 – 8115 = 166

  9. Question 9 of 27
    9. Question

    What is the least number to be added to 4321 to make it a perfect square?

    Hint

    number-system-q-49135.png

    ∴ (66)² – 4321 = 4356 – 4321 = 35

  10. Question 10 of 27
    10. Question

    What is the least number to be added to 4700 to make it a perfect square?

    Hint

    69 × 69 = 4761

    68 × 68 = 4624

    Clearly, 4624 < 4700 < 4761

    ∴ Hence, 61 should be added to make = 4761 – 4700 = 61

  11. Question 11 of 27
    11. Question

    What is the least number to be added to 3986 to make it a perfect square?

    Hint

    number-system-q-49129.png = 63.13

    ∴ Here, 63² = (64)² – 3986

    = 4096 – 3986 = 110

  12. Question 12 of 27
    12. Question

    What is the least number to be added to 1500 to make it a perfect square?

    Hint

    38² = 1444

    39² = 1521

    ∴ Required number

    = 1521 – 1500 = 21

  13. Question 13 of 27
    13. Question

    The difference between two numbers is 3 and the difference between their squares is 63. Which is the larger number?

    Hint

    Let the larger and smaller numbers be x and y respectively.

    Then, x – y = 3 …(i)

    and, x² – y² = 63

    ⇒ (x + y) (x – y) = 63

    ⇒ number-system-q-47974.png …(ii)

    From equation (i) and (ii),

    x = 12

  14. Question 14 of 27
    14. Question

    If (12)³ is subtracted from the square of a number the answer so obtained is 976. What is the number?

    Hint

    Let the number be x.

    x² – (12)3 = 976

    ∴ x² = 976 + 1728 = 2704

    ∴ x = number-system-q-47968.png

  15. Question 15 of 27
    15. Question

    If (56)² is added to the square of a number, the answer so obtained is 4985. What is the number?

    Hint

    Let the number be x.

    x² + (56)² = 4985

    ⇒ x² = 4985 – 3136 = 1849

    ∴ x = number-system-q-47962.png= 43

  16. Question 16 of 27
    16. Question

    The difference between a two-digit number and the number obtained by interchanging the two digits of the number is 9. The sum of the digits of the number is 15. What is the product of the two digits of the two-digit number?

    Hint

    Let the two-digit number be

    = 10 x + y, where x < y.

    Number obtained after interchanging the digits = 10 y + x

    According to the question,

    10 y + x – 10 x – y = 9

    ⇒ 9y – 9x = 9

    ⇒ 9(y –x) = 9

    ⇒ y – x = 1 …(i)

    and x + y = 15 …(ii)

    From equations (i) and (ii),

    y = 8 and x = 7

    ∴ Required product = 8 × 7 = 56

  17. Question 17 of 27
    17. Question

    The number obtained by interchanging the two digits of a two-digit number is less than the original number by 18. The sum of the two digits of the number is 16. What is the original number?

    Hint

    Let the number be (10x + y)

    Then, (10x + y) – (10y + x) = 18

    ⇒ 9x – 9y = 18

    ⇒ x – y = 2 …(i)

    and, x + y = 16 …(ii)

    ∴ x = 9, y = 7

    From equations (i) and (ii),

    So, the number is (10 × 9 + 7) = 97

  18. Question 18 of 27
    18. Question

    How many numbers from 1 to 100 are there each of which is not only exactly divisible by 4 but also has 4 as a digit?

    Hint

    The numbers from 1 to 100 which are exactly divisible by

    4 are 4, 8, 12, 16, 20,24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64, 68, 72, 76, 80, 84, 88, 92, 96, 100.

    But each number should have 4 as its digit.

    The required numbers are 4, 24, 40, 44, 48, 64, 84.

    Clearly, there are 7 such numbers.

  19. Question 19 of 27
    19. Question

    The numbers 1 to 29 are written side by side as follows 1234567891011 … 2829. If the number is divided by 9, then what is the remainder?

    Hint

    Sum of the digits of the ‘super’ number

    = 1 + 2 + 3 +…. + 29

    number-system-q-49353.png

    number-system-q-49347.png

    = number-system-q-49340.png

    Now, sum of digits in the number 435 = 4 + 3 + 5 = 12 which gives a remainder of 3 when divided by 9.

  20. Question 20 of 27
    20. Question

    If x959y is divisible by 44 and y >5, then what are values of the digit x and y?

    Hint

    Here 44 = 11 × 4

    ∴ the number must be divisible by 4 and 11 respectively. Test of 4 says that 9y must be divisible by 4 and since y > 5, so y = 6

    Again , x 9596 is divisible by 11,

    so x + 5 + 6 = 9 + 9

    ⇒ x = 7

    Thus x = 7, y = 6

  21. Question 21 of 27
    21. Question

    The quotient arising from the division of 24162 by a certain number x is 89 and the remainder is 43. Find x.

    Hint

    24162 = 89x + 43

    ⇒ x = (24162 – 43) ÷ 89 = 271

  22. Question 22 of 27
    22. Question

    Find the unit’s digit in the product (2467)¹⁵³ × (341)⁷².

    Hint

    Clearly, unit’s digit in the given product = unit’s digit in 7¹⁵³ × 1⁷².

    Now, 7⁴ gives unit digit 1.

    ∴ 7¹⁵³ gives unit digit (1 × 7) = 7. Also 1⁷² gives unit digit 1.

    Hence, unit’s digit in the product

    = (7 × 1) = 7.

  23. Question 23 of 27
    23. Question

    The unit’s digit in the product (7⁷¹×6⁵⁹×3⁶⁵) is:

    Hint

    Unit digit in 7⁴ is 1.

    Unit digit in 7⁶⁸ is 1.

    ∴ Unit digit in 7⁷¹ = 1 × 7³ = 3

    Again, every power of 6 will give unit digit 6.

    ∴ Unit digit in 6⁵⁹ is 6.

    Unit digit in 3⁴ is 1.

    ∴ Unit digit in 3⁶⁴ is 1.

    Unit digit in 3⁶⁵ is 3.

    ∴ Unit digit in (7⁷¹ × 6⁵⁹ × 3⁶⁵)

    = Unit digit in (3 × 6 × 3) = 4.

  24. Question 24 of 27
    24. Question

    How many times must 79 be subtracted from 5 × 10⁴ so as to obtain 43759?

    Hint

    Let x be the number of times, then

    79x + 43759 = 50,000

    ⇒ x = (50000 – 43759) ÷ 79 = 79

  25. Question 25 of 27
    25. Question

    55³ + 17³ – 72³ is divisible by

    Hint

    number-system-q-49334.png

    number-system-q-49328.png

    number-system-q-49322.png

    number-system-q-49316.png

  26. Question 26 of 27
    26. Question

    The digit in the unit’s place of the number represented by (7⁹⁵ – 3⁵⁸) is:

    Hint

    Unit digit in 7⁴ is 1. So, unit digit in 7⁹² is 1.

    ∴ Unit digit in 7⁹⁵ is 3.

    Unit digit in 3⁴ is 1.

    ∴ Unit digit in 3⁵⁶ is 1.

    ∴ Unit digit in 3⁵⁸ is 9.

    ∴ Unit digit in (7⁹⁵ – 3⁵⁸) = (13 – 9) = 4.

  27. Question 27 of 27
    27. Question

    What is the digit in the unit place of 2⁵¹?

    Hint

    The digit in the unit’s place of 2⁵¹ is equal to the remainder when 2⁵¹ is divided by 10. 2⁵ = 32 leaves the remainder 2 when divided by 10. Then 2⁵⁰ = (2⁵)¹⁰ leaves the remainder 2¹⁰ = (2⁵)² which in turn leaves the remainder 2² = 4. Then
    2⁵¹ = 2⁵⁰×2, when divided by 10, leaves the remainder 4×2 = 8.

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