How Many Numbers Between 300 and 900 Are Divisible by 17?

There are 35 numbers between 300 and 900 that are divisible by 17.

Analysis:

To determine the number of integers divisible by 17 within the specified range, we can use the concept of arithmetic progressions. The first step is to identify the first multiple of 17 greater than 300 and the last multiple of 17 less than 900.

• The first multiple of 17 above 300 is 306 (17 * 18).
• The last multiple of 17 below 900 is 884 (17 * 52).

We can now see that the multiples of 17 within the range form an arithmetic progression (AP) with:

• First term (a) = 306
• Common difference (d) = 17
• Last term (l) = 884

To find the number of terms (n) in this AP, we can use the formula:

l = a + (n - 1) * d

Substituting the values:

884 = 306 + (n - 1) * 17

Now, solve for 'n':

884 - 306 = (n - 1) * 17
578 = (n - 1) * 17
578 / 17 = n - 1
34 = n - 1
n = 34 + 1
n = 35

Therefore, there are 35 multiples of 17 between 300 and 900. The provided reference confirms this calculation, stating, "We have an AP whose first term is 306 and the last term is 884 and the cd = 17. n = 52–18+1 = 35. There are 35 terms between 300 and 900 that are divisible by 17."