Counting & Summation - Atreya Roy


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    Author: Atreya Roy is pursuing his BTech From Kalyani Government Engineering College, Bengal.

    Many a time we face questions where we are told to find the total number of digits used in writing numbers or digits used in pages of books. Today we will learn how to do these kind of sums.

    Number of Total Digits of all natural numbers starting from 1 to any given value.

    Number of single digits = 1 to 9 = 9

    Number of 2 digits = 10 to 99 = 90 numbers.

    90 2-digit numbers = 2*90= 180 digits.

    3 digits numbers = 100 to 999 = 900 . = 900*3 = 2700 digits.

    Generally sums are till 3 digits only, hence we will just to upto that.

    Calculate the number of digits present in the total number that has been created by writing all natural numbers till 250

    Solution :
    1 to 9 = 9 digits
    10 to 99 = 90 * 2 = 180 digits
    250 to 100 = 151 * 3 = 453 digits
    Hence total number of digits = 642.

    Note : USE this concept for the reverse way too. Suppose the total number of digits are given and you need to find out the last 3 or 4 digits. We will use this formula.

    A number is formed by writing all natural numbers. The number contains 1000 digits. What is the remainder if this number is divisible by 8 ?

    Here we need two concepts :

    • Calculate the last 3 digits by the rule we discussed above
    • Divisibility by 8 = div of the last 3 digits by 8

    Hence. We will try to figure out the last 3 digits.
    1 to 9 = 9 digits
    10 – 99 = 90 * 2 = 180
    Digits remaining = 1000-189 = 811
    We see 810/3 = 270. Hence our Second last number is 100+269 = 369
    Next number = 370. But we will only take the first digit since our 1000 digits are complete there.
    So last 3 digits = 693.
    Now applying the divisibility rule of 8, 693 mod 8 = 5

    Concept of pages in a book :

    Before we learn this concept : we need the formula of the sum of first “n” naturals since, book pages always start with 1 .

    Sum = n(n+1)/2

    We will be given a sum, where total sum of the page numbers will be given and some pages are torn. We need to identify those pages which are torn. For these we need to calculate the sum of the page numbers just exceeding the given value. (Just the next value). Then , we will subtract the given sum from this. The difference is always the sum of the page numbers.

    If the difference is say 25. There may be 2 cases :

    1. Page 25 is torn
    2. Page 12 and 13 is torn.

    The first case is wrong, because in a book. The pages are back to back. Tearing 25th page means either of 24 or 26 is also torn. Hence we cannot reach a conclusion here. So our answer will be 12th and 13th page. But in case of Inkloss and the page number has been erased, keep in mind, only one page (25th page) is to be considered, since inkloss is in a single page (generally). So read the question properly.

    The sum of page numbers of the book “BlackHawk” is 1000. It was observed that a page is torn off from the book. Which page was it ?

    Sum = 1000
    N(n+1)/2 = 1000+ (Since we need to calculate the next upper value)
    45 * (46) /2 = 1035.
    Hence Sum of pages torn = 35
    Thus, Page 17 and 18 are torn.


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