To determine how many four-digit numbers formed by using the digits 1,2,3, and 5 without repetition are divisible by 4 , we need to understand the divisibility rule for 4 . A number is divisible by 4 if the last two digits of the number form a number itself divisible by 4 . Let's examine which pairs of digits from 1,2,3, and 5 result in a number divisible by 4 : 12(‌ since ‌12÷4=3) 32(‌ since ‌32÷4=8) 52(‌ since ‌52÷4=13‌ ) ‌ So, the valid pairs for the last two digits are (1,2),(3,2), and (5,2). We now need to form four-digit numbers where the last two digits are one of these pairs. For each pair of last two digits, we can arrange the remaining two digits (chosen from the remaining digits) in the first two positions. Let's calculate the possibilities: If the last two digits are 12 : The remaining two digits are 3 and 5 . These can be arranged in 2 ! ( 2 factorial) ways. 2!=2×1=2 If the last two digits are 32 : The remaining two digits are 1 and 5 . These can be arranged in 2 ! ways. 2!=2×1=2 If the last two digits are 52 : The remaining two digits are 1 and 3 . These can be arranged in 2 ! ways. 2!=2×1=2 Since we have three valid pairs of last two digits, and each pair offers 2 valid arrangements of the remaining digits, the total number of four-digit numbers divisible by 4 is: 3×2=6