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How many 4 digit numbers greater than 4000 can be formed using the digits from 0 to 8 such that the number is divisible by 4?
  • a)
    508
  • b)
    827
  • c)
    828
  • d)
    1034
  • e)
    1035
Correct answer is option 'D'. Can you explain this answer?
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How many 4 digit numbers greater than 4000 can be formed using the dig...
Given
  • Digits to be used → 0 – 8
    • Digits can be repeated
  • Numbers to be divisible by 4
To Find: 4 digit integers > 4000 and divisible by 4?
Approach
  1. We need to find the number of 4 digit integers greater than 4000 that are divisible by 4.
  2. For an integer to be divisible by 4, the last two digits should be divisible by 4. So, we can divide this into the following tasks:
  3. Task-1: Selecting the digit at thousands place
    1. The thousands place can have a digit ≥ 4 and ≤ 8
  4. Task-2: Selecting the digit at the hundreds place
    1. The hundreds place can take any digit from 0-8
  5. Task-3: Selecting the last 2 digits
    1. So, we need to find the number of 2 digit integers that are divisible by 4 such that none of them contains a 9, i.e. we need to find the number of 2 digit integers divisible by 4 between 00 to 88, inclusive.
      1. Please note that we can take 00 as one of the possibilities because the last 2 digits are a part of the 4 digit integer.
  6. However, as we have taken 4 as one of the thousands place digit, there can be a case where the integer can be equal to 4000. So, we need to subtract this possibility when calculating our final answer.
  7. So, number of 4 digit integers greater than 4000 = Task-1 * Task-2 * Task -3 – 1
 
Working Out
  1. Task-1: Selecting the digit at thousands place
    1. Number of ways to select the digit at thousands place = {4, 5, 6, 7, 8}, i.e. 5 ways..(1)
  2. Task-2: Selecting the digit at hundreds place
    1. Number of ways to select the digit at hundreds place = {0- 8) , i.e. 9 ways…..(2)
  3. Task-3: Selecting the last 2 digits:
    1. Number of multiples of 4 from 00 to 88, i.e. from 4*0 to 4*22 = 22 + 1 = 23
  4. Number of 4 digit integers greater than 4000 = 5 * 9 * 23 – 1 = 1035 – 1 = 1034
 
Answer: D
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Directions: Read the given passage carefully and answer the question as follow.Among the speculative questions which arise in connection with the study of arithmetic from a historical standpoint, the origin of number is one that has provoked much lively discussion, and has led to a great amount of learned research among the primitive and savage languages of the human race. A few simple considerations will, however, show that such research must necessarily leave this question entirely unsettled, and will indicate clearly that it is, from the very nature of things, a question to which no definite and final answer can be given. Among the barbarous tribes whose languages have been studied, even in a most cursory manner, none have ever been discovered which did not show some familiarity with the number concept. The knowledge thus indicated has often proved to be most limited; not extending beyond the numbers 1 and 2, or 1, 2, and 3. At first thought it seems quite inconceivable that any human being should be destitute of the power of counting beyond 2. But such is the case; and in a few instances languages have been found to be absolutely destitute of pure numeral words.These facts must of necessity deter the mathematician from seeking to push his investigation too far back toward the very origin of number. Philosophers have endeavoured to establish certain propositions concerning this subject, but, as might have been expected, have failed to reach any common ground of agreement. Whewell has maintained that “such propositions as that two and three make five are necessary truths, containing in them an element of certainty beyond that which mere experience can give.” Mill, on the other hand, argues that any such statement merely expresses a truth derived from early and constant experience; and in this view he is heartily supported by Tylor.But why this question should provoke controversy, it is difficult for the mathematician to understand. Either view would seem to be correct, according to the standpoint from which the question is approached. We know of no language in which the suggestion of number does not appear, and we must admit that the words which give expression to the number sense would be among the early words to be formed in any language. They express ideas which are, at first, wholly concrete, which are of the greatest possible simplicity, and which seem in many ways to be clearly understood, even by the higher orders of the brute creation. The origin of number would in itself, then, appear to lie beyond the proper limits of inquiry; and the primitive conception of number to be fundamental with human thought.Q.What does the line, in the third para, ‘primitive conception of number to be fundamental with human thought’ mean?

Directions: Read the given passage carefully and answer the question as follow.Among the speculative questions which arise in connection with the study of arithmetic from a historical standpoint, the origin of number is one that has provoked much lively discussion, and has led to a great amount of learned research among the primitive and savage languages of the human race. A few simple considerations will, however, show that such research must necessarily leave this question entirely unsettled, and will indicate clearly that it is, from the very nature of things, a question to which no definite and final answer can be given. Among the barbarous tribes whose languages have been studied, even in a most cursory manner, none have ever been discovered which did not show some familiarity with the number concept. The knowledge thus indicated has often proved to be most limited; not extending beyond the numbers 1 and 2, or 1, 2, and 3. At first thought it seems quite inconceivable that any human being should be destitute of the power of counting beyond 2. But such is the case; and in a few instances languages have been found to be absolutely destitute of pure numeral words.These facts must of necessity deter the mathematician from seeking to push his investigation too far back toward the very origin of number. Philosophers have endeavoured to establish certain propositions concerning this subject, but, as might have been expected, have failed to reach any common ground of agreement. Whewell has maintained that “such propositions as that two and three make five are necessary truths, containing in them an element of certainty beyond that which mere experience can give.” Mill, on the other hand, argues that any such statement merely expresses a truth derived from early and constant experience; and in this view he is heartily supported by Tylor.But why this question should provoke controversy, it is difficult for the mathematician to understand. Either view would seem to be correct, according to the standpoint from which the question is approached. We know of no language in which the suggestion of number does not appear, and we must admit that the words which give expression to the number sense would be among the early words to be formed in any language. They express ideas which are, at first, wholly concrete, which are of the greatest possible simplicity, and which seem in many ways to be clearly understood, even by the higher orders of the brute creation. The origin of number would in itself, then, appear to lie beyond the proper limits of inquiry; and the primitive conception of number to be fundamental with human thought.Q.What is the primary purpose of the passage?

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How many 4 digit numbers greater than 4000 can be formed using the digits from 0 to 8 such that the number is divisible by 4?a)508b)827c)828d)1034e)1035Correct answer is option 'D'. Can you explain this answer?
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