(i) 71640, 321970
(ii) 1143, 47205, 2316
(iii) 1428, 9252, 7212
(iv) 24620, 9100, 6700
(v) 12320, 59016, 46424
Step-by-step explanation:
(i) In order for any number to be divisible by 5, its last digit must be either 0 or 5.
In this case, we should add a last digit, so we opt for 0 without any doubt since it gives a smaller number than 5 as the last digit: 71640 is smaller than 71645, the same way as 321970 is smaller than our second option, 321975.
(ii) If the sum of a number's digits gives a number divisible by 3, then the main number is also divisible by 3.
This means that we have to do some addition:
1_43: 1 + 4 + 3 = 8. Eight is not divisible by 3. But 9 is. So the smallest digit we can add for it to be divisible by three is 1: 1143.
47_05: 4 + 7 + 0 + 5 = 16. Since 16 is not, the next number that is divisible by 3 is 18, and therefore, the smallest digit we need here is 2: 47205.
_316: 3 + 1 + 6 = 10. Twelve is the next number divisible by 3, and that means 2 is the digit we choose: 2316.
(iii) Only numbers that are divisible by both 2 and 3 are also divisible by 6.
_428: In order for a number to be divisible by 2, its last digit must be 0 or divisible by 2. That is already the case here.
4 + 2 + 8 = 14 which is not divisible by 3. But 15 is. So the smallest digit we need is 1: 1428.
9_52: The number will be divisible by 2. And 9 + 5 + 2 = 16 which is not divisible by 3 but 18 is. This means the smallest digit we need is 2: 9252.
721_: Since the last digit can be any of these 0, 2, 4, 6, 8 for the number to be divisible by two, we will first see what digit we need for it to be divisible by 3.
7 + 2 + 1 = 10 which is not divisible by 3 but 12 is. We will then choose to add 2: 7212.
(iv) A number is divisible by 4 only if the number that is formed by its two last digits is also divisible by 4..
2462_: Since we need to add a last digit, this should be easy. The smallest two-digit-number that starts with 2 and is divisible by four is 20: 24620.
91__: Here we need to add both of the two last digits, and this should be the easiest example! All numbers that end in 00, meaning they have a round number of hundreds, are divisible by 4! Therefore our option is: 9100.
670_: In this case, we apply the same rule from the last example - and we choose 0: 6700.
(v) In order for a number to be divisible by 8, its last three digits need to be divisible by 8, or it can end in 000, since every number having a round number of thousands is also divisible by 8.
1232_: 320 is divisible by 8 since 8 × 40 gives 320. Therefore, our smallest digit here is 0: 12320.
59_16: If we add a 0 here, the last three digits form the number 016, which is divisible by 8. Our choice is again zero: 59016.
4642_: Unfortunately, 420 is not divisible by 8. The next number closest to 420 and which is divisible by 8 is 424. Its division by 8 gives 53. Therefore we should add 4: 46424.
