Question

Is 4320 perfect, abundant, or deficient? also perfect numbers? Explain why all positive multiples of 6 greater than 6 are aburndant numbers.

Answer 1

The number 4320 is an abundant number.

Explanation:

To find : Is 4320 is perfect, abundant, or deficient? Explain why all positive multiples of 6 greater than 6 are abundant numbers.

Solution :

First we define all terms,

1) A perfect number is one whose factors are equal to a given number i.e. P(n)=n

2) An abundant number is a composite number whose factors, without the number itself, have a sum greater than the number i.e. P(n) >n

3) A deficient number is a composite number in which the sum of its factors is less than the given number i.e. P(n) < n

We have given number - 4320

Factors of 4320 are 1,2,3,4,5,6,8,9,10,12,15,16,18,20,24,27,30,32,36,40,45,48,54,60,72,80,90,96,108,120,135,144,160,180,216,240,270,288,360,432,480,540,720,864,1080,1440,2160,4320.

If we don't take 4320 as a factor,

Then the sum is

`S=1+2+3+4+5+6+8+9+10+12+15+16+18+20+24+27+30+32+36+40+45+48+54+60+72+80+90+96+108+120+135+144+160+180+216+240+270+288+360+432+480+540+720+864+1080+1440+2160\\\\S=10800`

Now, 10800 >4320

Therefore, The number 4320 is an abundant number.

If we take positive multiples of 6,

Factors of 6, 1,2,3 and 6

The sum of factors except 6,

1+2+3=6

Which means Multiples of 6 is a perfect number.

So, Greater than 6 are abundant numbers as their sum is greater.