Let Sm denote the sum of the squares of first m natural numbers. For How many values of m<100,is Sm a multiple of 4?

1) 50
2) 25
3) 36
4) 24

• 24
  Sm=(1^2 +2^2+....+m^2)=m(m+1)(2m+1)/6,
  if the sum is divisible by 4 then ,we can say that " m(m+1)(2m+1)/24 " will be an integer.
  from (1-10) => 7,8
  (11-20)=>15,16
  ..........................
  (91-99) => 95,96
  so here we get an AP i.e.
  7,8,15,16.......95.96. => 7,15,.... ,95 && 8,16, ...... , 96
  number of terms in both series are,
  12+12=24
• 10 years agoHelpfull: Yes(1) No(1)
• its 24
  Sm=(1^2 +2^2+....+m^2)=m(m+1)(2m+1)/6
• 10 years agoHelpfull: Yes(0) No(0)