Question

Apositive integer n is called special if n is divisible by 4. n +1 is divisible by 5, and n + 2 is divisible by 6. How many special integers smaller than 1000 are there,​

Answer 1

17

Explanation:

these positive integers are :

4, 64, 124, 184 ... , 964

964 = 4 + (n-1) (60)

964 = 4 + 60n -60

964 = 60n -56

60n = 1020

n = 1020/60

n = 17

so, the number of special integers smaller than 1000 are = 17

Answer 2

Answer: 17

Hello,

Explanation:

n is divisible by 4.

n +1 is divisible by 5

n + 2 is divisible by 6

n <1000

`\left\{\begin{array}{ccc}n&=&4*k_1\\n+1&=&5*k_2\\n+2&=&6*k_3\\\end{array}\right.\\\\4*k_1+1=5k_2\Longrightarrow\ k_2=(4k_1+1)/(5) \Longrightarrow\ k_1=1+5*t\ , k_2=1+4*t\\\\4*k_1+2=6k_3\Longrightarrow\ k_3=(4(1+5t)+2)/(6) \Longrightarrow\ k_3=1+(10*t)/(3) \\\\\Longrightarrow\ t=3z\\k_3=1+10z\\n+2=6(1+10z)\\n=4+60z\\\\1000 < 4+60z \Longrightarrow\ z \leq 16\\`

There are 17: from 4+0*60=4 to 4+16*60=964.