How many numbers amongst the numbers 5 to 141 are there which are exactly…

Concept: The count of multiples of n up to a limit L is floor(L/n). When you need numbers divisible by one number (9) but NOT by another (4), first count all multiples of 9 in the range, then subtract those multiples of 9 that are ALSO multiples of 4 — these are exactly the multiples of LCM(9, 4) = 36.

1. Count multiples of 9 between 5 and 141: the smallest is 9×1 = 9 and the largest is 9×15 = 135 (9×16 = 144 exceeds 141), giving 15 multiples of 9.

2. Find LCM(9, 4) = 36 — a number divisible by both 9 and 4 must be a multiple of 36.

3. Count multiples of 36 between 5 and 141: 36×1 = 36, 36×2 = 72, 36×3 = 108 (36×4 = 144 exceeds 141), giving 3 multiples of 36.

4. Subtract the common multiples from the total: 15 − 3 = 12 numbers are divisible by 9 but not by 4.

Cross-check: listing the multiples of 9 (9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99, 108, 117, 126, 135) and removing 36, 72, 108 leaves exactly 12 values, confirming the count.