Question Bank  Number theory  Sibanand Pattnaik

Q3) How many factors of 2^3 * 3^2 * 5^4 are multiples of 5 ?

Q4) Number of ways in which 2^3 * 3^2 * 5^4 be resolved into product of 2 coprimes (a,b,) ?

Q5) How many factors of 2^3 * 3^2 * 5^4 are less than and coprime to it ? Also, find the sum of all those factors.

Q6) In how many ways can 2^3 * 3^2 * 5^4 be written as SUM of consecutive natural numbers and as SUM of consecutive integers ?

Q7) How many pairs exist which have LCM = 2^3 * 3^2 * 5^4 ?

Q8) How many triplets exist which have LCM = 2^3 * 3^2 * 5^4 ?

Q9) How many Factor triplets (a,b,c) of 2^3 * 3^2 * 5^4 exist which have HCF = 1

Q10) Number of ways in which 2^3 * 3^2 * 5^4 can be resolved into product of 3 coprimes? Also, find the number of ways the 2^3 * 3^2 * 5^4 can be written as the SUM of 2 coprimes.

Q11) Find the sum of all 4 digit numbers such that its Thousands digit Tens digit and Units digits are in ratio 2:3 :1 and hundredth digit is equal to units digit ?

Q12) Out of first 20 natural numbers , how many 3 element subsets can be formed such that the product of 3 elements is divisible by 4 ?

Q13) How many ways can 720 be expressed as Product of 2 co primes

Q14) If 1/a + 1/b + 1/c + 1/d = 2, where a , b , c , d are distinct natural numbers. what is the value of a + b + c + d ?

Q15) If (1+2^x) is a divisor of 1 + 2 + 2^2 +2^3 +2^4 +........+ 2^255 then what is the max value of "x" possible ?

Q16) How many distinct digits are there in the largest 12 digit perfect square ?

See there is a pattern the series of Squares of 9's
9^2 = 81
99^2 = 9801
999^2 = 998001
9999^2 = 99980001
Like wise 12 digit is 999998000001 which is 999999^2
this is pattern . i hope u r able to relate to this ..
so how many distinct digits are there in 999998000001 ??
its 4

Q17) How many distinct pair of factors of 21000 have HCF as 75 ?

Q18) The HCF of x, y and z is 8 and the LCM of x,y, z is a 4 digit number .. if xy = yz= 8 .. find the maximum possible value of z

Q19) Soham enters all the natural numbers from 1 to k (k < 100), in ascending order, in the first column of a spreadsheet, without leaving any cells blank. He then enters the same numbers from k to 1, in descending order, in the second column of the same sheet. He then deletes all the rows in which one entry is a factor of the other. He has to delete 8 rows. How many rows were there initially?
a) 32
b) 16
c) 23
d) 80

Q20) In a 4 digit no. having non zero and distinct digits, the sums of the digits at the unit place and the tens place is equal to the sum of other 2 digits. The sum of the digits at the tens and the hundreds places is three times the sum of the remaining 2 digits. If the sum of the digits is atmost 20, then how many such 4 digit numbers are possible ?

Q21) Sum of digits of an integer N is taken, called digital sum of the integer N. (for eg. digital sum of 12345 = 1 + 2 + 3 + 4 + 5 = 15). Suppose x, y and z are three integers with the same number of digits such that x + y = z. If the digital sum of x is p and that of y is q, then what is the digital sum of z given that there were exactly n ‘carries’ when the addition is performed ?
a. p + q  7n
b. p  q + n
c. p + q  9n
d. p + q + 2n