2IIM Quant Notes  Number Theory and Counting

rajesh_balasubramanian
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. 12Two digit numbers; The two digits can be 4 and 7: Two possibilities 47 and 74
Threedigit 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 5digit 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 5digit 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 nonzero 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 nonzero units digit = 20057 = 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 3digit number. This will be the case for numbers from 3741, 7377, 109113 and 145149. 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. 3780We need to consider all three digit and all 4digit numbers.
Threedigit 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 3digit numbersFourdigit 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 4digit numbers.So, there are totally 3700 + 280 = 3980 numbers