Number of divisors and Integer | B.Stat Objective | TOMATO 83

Try this TOMATO problem from I.S.I. B.Stat Entrance Objective Problem based on Number of divisors and Integer.

Number of divisors and Integer (B.Stat Objective)

The smallest positive integer n with 24 divisors (where 1 and n are also considered divisors of n) is

Key Concepts

Number of divisors

Integer

Least positive integer

Check the Answer

Answer: 360

B.Stat Objective Question 83

Challenges and Thrills of Pre-College Mathematics by University Press

Try with Hints

number of divisors of 420=(2+1)(1+1)(1+1)(1+1)=24

number of divisors of 240=(4+1)(1+1)(1+1)=20

number of divisors of 360=(3+1)(2+1)(1+1)=24 and number of divisors of 480=(5+1)(1+1)(1+1)=24 then required number =360.

Least Positive Integer Problem | AIME I, 2000 | Question 1

Try this beautiful problem from the American Invitational Mathematics Examination, AIME, 2000 based on Least Positive Integer.

Least Positive Integer Problem - AIME I, 2000

Find the least positive integer n such that no matter how \(10^{n}\) is expressed as the product of any two positive integers, at least one of these two integers contains the digit 0.

is 107
is 8
is 840
cannot be determined from the given information

Key Concepts

Product

Least positive integer

Integers

Check the Answer

Answer: is 8.

AIME I, 2000, Question 1

Elementary Number Theory by Sierpinsky

Try with Hints

\(10^{n}\) has factor 2 and 5

for n=1 \(2^{1}\)=2 \(5^{1}\)=5

for n=2 \(2^{2}\)=4 \(5^{2}=25\)

for n=3 \(2^{3}\)=8 \(5^{3}=125\)

........

for n=8 \(2^{8}\)=256 \(5^{8}=390625\)

here \(5^{8}\) contains the zero then n=8.

Number of divisors and Integer | B.Stat Objective | TOMATO 83