How many whole numbers are there between 244 and 332 which are exactly…

Concept

To count multiples of a number d lying strictly between two integers, first identify the smallest multiple of d greater than the lower bound and the largest multiple of d less than the upper bound. These multiples form an arithmetic progression (AP) with common difference d, so the number of terms between them can be found with the AP formula an = a + (n − 1)d.

Application

1. 244 ÷ 7 = 34 remainder 6, so the smallest multiple of 7 greater than 244 is 7 × 35 = 245.

2. 332 ÷ 7 = 47 remainder 3, so the largest multiple of 7 less than 332 is 7 × 47 = 329.

3. The multiples 245, 252, 259, ..., 329 form an AP with first term a = 245, common difference d = 7, and last term an = 329.

4. Using an = a + (n − 1)d: 329 = 245 + (n − 1) × 7 ⇒ (n − 1) × 7 = 84 ⇒ n − 1 = 12 ⇒ n = 13.

Cross-check

Counting multiples of 7 up to each bound independently gives ⌊332 ÷ 7⌋ = 47 and ⌊244 ÷ 7⌋ = 34; the difference 47 − 34 = 13 confirms the same count.

∴ There are 13 whole numbers between 244 and 332 that are exactly divisible by 7.