2IIM Quant Notes - Number Theory and Counting


  • Director, 2IIM Online CAT Preparation | IIT Madras | IIM Bangalore | CAT 100th percentile - CAT 2011, 2012 and 2014.


    CAT has been consistently asking questions combining basic number theory and counting. So, it is probably good practice to have a go at these.

    How many numbers with distinct digits are possible product of whose digits is 28?
    A. 6
    B. 4
    C. 8
    D. 12

    Two digit numbers; The two digits can be 4 and 7: Two possibilities 47 and 74
    Three-digit numbers: The three digits can be 1, 4 and 7: 3! Or 6 possibilities.
    We cannot have three digits as (2, 2, 7) as the digits have to be distinct.
    We cannot have numbers with 4 digits or more without repeating the digits.
    So, there are totally 8 numbers.

    From the digits 2,3,4,5,6 and 7, how many 5-digit numbers can be formed that have distinct digits and are multiples of 12?

    Any multiple of 12 should be a multiple of 4 and 3. First, let us look at the constraint for a number being a multiple of 3. Sum of the digits should be a multiple of 3. Sum of all numbers from 2 to 7 is 27. So, if we have to drop a digit and still retain a multiple of 3, we should drop either 3 or 6.

    So, the possible 5 digits are 2, 4, 5, 6, 7 or 2, 3, 4, 5, 7.

    When the digits are 2, 4, 5, 6, 7. the last two digits possible for the number to be a multiple of 4 are 24, 64, 52, 72, 56, 76. For each of these combinations, there are 6 different numbers possible. So, with this set of 5 digits we can have 36 different numbers

    When the digits are 2, 3, 4, 5, 7. the last two digits possible for the number to be a multiple of 4 are 32, 52, 72, 24. For each of these combinations, there are 6 different numbers possible. So, with this set of 5 digits we can have 24 different numbers

    Overall, there are 60 different 5-digit numbers possible

    All numbers from 1 to 200 (in decimal system) are written in base 6 and base 7 systems. How many of the numbers will have a non-zero units digit in both base 6 and base 7 notations?

    If a number written in base 6 ends with a zero, it should be a multiple of 6. In other words, the question wants us to find all numbers from 1 to 200 that are not multiples of 6 or 7. There are 33 multiples of 6 less than 201. There are 28 multiples of 7 less than 201. There are 4 multiples of 6 & 7 (or multiple of 42) from 1 to 200.

    So, total multiples of 6 or 7 less than 201 = 33 + 28 – 4 = 57. Number of numbers with non-zero units digit = 200-57 = 143.

    All numbers from 1 to 150 (in decimal system) are written in base 6 notation. How many of these will not contain any zero?

    Any multiple of 6 will end in a zero. There are 25 such numbers. Beyond this, we can have zero as the middle digit of a 3-digit number. This will be the case for numbers from 37-41, 73-77, 109-113 and 145-149. There are 20 such numbers. Overall, there are 45 numbers that have a zero in them.

    How many factors of 1080 are perfect squares?

    1080 = 2^3 * 3^3 * 5. For any perfect square, all the powers of the primes have to be even numbers. So, if the factor is of the form 2^a * 3^b * 5^c. The values ‘a’ can take are 0 and 2, b can take are 0 and 2, and c can take the value 0. Totally there are 4 possibilities. 1, 4, 9, and 36.

    This is an interesting question from Counting. Simple framework, but one needs to be very careful with the enumeration. One can get wrong answers in a number of ways.

    If we listed all numbers from 100 to 10,000, how many times would the digit 3 be printed?
    A. 3980
    B. 3700
    C. 3840
    D. 3780

    We need to consider all three digit and all 4-digit numbers.

    Three-digit numbers: A B C. 3 can be printed in the 100’s place or10’s place or units place.
    Ø 100’s place: 3 B C. B can take values 0 to 9, C can take values 0 to 9. So, 3 gets printed in the 100’s place 100 times
    Ø 10’s place: A 3 C. A can take values 1 to 9, C can take values 0 to 9. So, 3 gets printed in the 10’s place 90 times
    Ø Unit’s place: A B 3. A can take values 1 to 9, B can take values 0 to 9. So, 3 gets printed in the unit’s place 90 times
    So, 3 gets printed 280 times in 3-digit numbers

    Four-digit numbers: A B C D. 3 can be printed in the 1000’s place, 100’s place or10’s place or units place.
    Ø 1000’s place: 3 B C D. B can take values 0 to 9, C can take values 0 to 9, D can take values 0 to 9. So, 3 gets printed in the 100’s place 1000 times.
    Ø 100’s place: A 3 C D. A can take values 1 to 9, C & D can take values 0 to 9. So, 3 gets printed in the 100’s place 900 times.
    Ø 10’s place: A B 3 D. A can take values 1 to 9, B & D can take values 0 to 9. So, 3 gets printed in the 10’s place 900 times.
    Ø Unit’s place: A B C 3. A can take values 1 to 9, B & C can take values 0 to 9. So, 3 gets printed in the unit’s place 900 times.
    3 gets printed 3700 times in 4-digit numbers.

    So, there are totally 3700 + 280 = 3980 numbers


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