How many positive integers ‘n’ less than 17 are there for which n! + (n+1)! + (n+2)! is an integral multiple of 49?

1) 0
2) 1
3) 3
4) 5

• Factor. n! + (n+1)! + (n+2)! = n!(1+n+1+(n+1)(n+1))=n!(n+2)^2. Thus the quantity in question is divisible by 49 if and only if n+2 is divisible by 7 or n! is divisible by 49. The first condition happens if and only if n=5 or n=12, and the second condition happens if and only if n>=14. Thus the integers we seek are {5, 12, 14, 15,16} So 4) 5 integers
• 8 years agoHelpfull: Yes(1) No(0)