Factors

Puzzle

There are a 1000 locks in closed position. 1000 children numbering 1 to 1000 approach the lock one by one. The rule is the child (say with number 'n') can open or close the lock numbering  in multiples of "n" only.
so the child in number one position can open all the locks. Child number 2 will be able to close all the locks in multiples of 2 (like 2, 4, 6 .... upto 1000). The other locks will remain as they are.
After all the 1000 children have had their chance, how many locks will be in open position.

factors

How many factors of 7¹⁴ x 8⁷ x 10⁵ are a multiple of 160?

factors

How many factors are there for the number 264?

factors

How many even factors are there for the number 6¹² x 7¹⁴ x 8⁷ x 10⁵

factors and number

If the sum of the factors of a number is 124. What is the number?

factors

For the number 6¹⁰ x 7¹⁷ x 55²⁷, find

1.       The number of prime factors

2.       The number of all the factors.

tutorials -sum of factors

Sum of Factors

If a number N can be expressed as product of prime factors such that

N = p^a x q^b x r^c...............................

Write the power of each of the prime factors beginning with zero and going to thee highest power of the prime number.

Record the same for each prime number.

An example will help

64= 2⁶

Then the sum of factors (2⁰ + 2¹ + 2² + 2³ + 2⁴ + 2⁵ + 2⁶) = 1 + 2 + 4 + 8 + 16 + 32 + 64 =  127

Another Example:

Find the total number of factors of 222.

222 = 2 x 3 x 37

Or, 222 = 2¹ x 3¹ x 37¹

then the sum of factors are (2⁰ + 2¹) (3⁰ + 3¹) (37⁰ + 37¹) = (1+2)(1+3)(1+37) = 3 x 4 x 38 = 456

We will check the answer

The factors of 222 are 1, 2, 3, 6, 37, 74, 111, 222.

Sum of factors = 456

tutorial on number of factors

To find the number of factors of a number

If a number N can be expressed as product of prime factors such that

N = p^a x q^b x r^c...............................

Then, the total number of factors including N and 1 is given by

(a + 1)x(b + 1)x(c + 1).................................

Example:

Find the total number of factors of 222.

222 = 2 x 3 x 37

Or, 222 = 2¹ x 3¹ x 37¹

So, the total number of factors = (1+1)x(1+1)x(1+1) = 2x2x2 = 8

We can check the answer as shown below.

The factors of 222 are 1, 2, 3, 6, 37, 74, 111, 222.

Find the total number of factors of 64.

64 = 2⁶

So, the total number of factors = (6+1) = 7

We can check the answer as shown below.

The factors of 64 are 1, 2, 4, 8, 16, 32, 64.

JMO

The least common multiple of two natural numbers a and b, is 399. What is the minimum possible sum of the digits of the number a(given a > b)?

Grouping in factors for STD V

Grouping in factors for STD V

solution to question paper 1 - IPM STD V

The least number to be multiplied by 17640 so that the result becomes a perfect square is -------------

Answer:

17640 = 3 x 3 x 2 x 2 x 7 x 7 x 2 x 5

So, the least that should be multiplied = 2 x 5 = 10 to make it a perfect square.

factors trick 2

Say you have a number 64.
It can be written as 2 x 2 x 2 x 2 x 2 x 2 x 1.
The no. of factor will simply be the count of all the factors.
ie. 7

number of factors trick 1

consider the number 210
it can be written as 210 = 2 x 5 x 3 x 7
All the factors are prime, and all the prime factors occur only once.
in such a case, the total number of factors is 2 x 2 x 2 x 2. (as many 2s' as there are prime factors).
So, the answer is 16.

consider 105.
it can be written as 105 = 3 x 5 x 7
All the factors are prime, and all the prime factors occur only once.
in such a case, the total number of factors is 2 x 2 x 2  (as many 2s' as there are prime factors).
So, the answer is 8.

listing the factors