Which digits should come in the place of * and #, respectively, if the 7-digit number 62684*# is divisible by both 8 and 5?
- (a)0 and 5
- (b)4 and 0
- (c)5 and 0
- (d)2 and 0
Answer
Why
Correct — B. Apply the two divisibility rules in order, starting with the one that pins a digit down.
Divisible by 5 means the last digit # is 0 or 5.
Divisible by 8 means the number is even, which rules out 5, so # = 0.
Divisibility by 8 also requires the last three digits to be a multiple of 8. Those digits are 4, then the starred digit, then 0.
440 ÷ 8 = 55, a whole number.
So the starred digit is 4 and # is 0, giving 6268440 → option (b)
Why the others are wrong
- (a)0 and 5 — 0 and 5 falls at the first hurdle. Ending in 5 makes 6268405 an odd number, and no odd number is divisible by 8, so 5 cannot stand once 8 is also required.
- (c)5 and 0 — 5 and 0 clears the divisible-by-5 test but not the other: the last three digits are 450, and 450 ÷ 8 = 56.25.
- (d)2 and 0 — 2 and 0 also ends correctly but leaves last three digits of 420, and 420 ÷ 8 = 52.5. Only 400, 440 and 480 work in that slot.
Concept
When a number must satisfy two divisibility rules at once, apply the more restrictive rule to whichever digit it fixes first.
Divisibility by 5 narrows the last digit to 0 or 5. Divisibility by 8 requires an even number, which kills the 5 and leaves 0.
The rule for 8 is that the last three digits form a multiple of 8. Everything from the thousands place up is a multiple of 1000, and 1000 is itself divisible by 8, so the leading 6268 can be ignored entirely.
With the last digit fixed at 0, the starred digit could be 0, 4 or 8, since 400, 440 and 480 are all multiples of 8. The option list offers only the 4, so the item has a single answer even though the arithmetic has three.
Key facts
- A number is divisible by 8 when its last three digits form a multiple of 8, because 1000 is divisible by 8.
- A number is divisible by 5 when it ends in 0 or 5, and by 10 only when it ends in 0.
- 440 ÷ 8 = 55 exactly, while 450 ÷ 8 = 56.25 and 420 ÷ 8 = 52.5.
Study next
Common traps
- Testing divisibility by 8 on the last two digits, which is the rule for 4
- Accepting a final 5 because it satisfies the rule for 5, without checking that 8 needs an even number
- Dividing the whole seven-digit number when only its last three digits matter
The item gives two conditions and two blanks, and the fast route is to let the stricter condition fix a digit first — divisibility by 8 settles the last digit before you look at the starred one at all.
Write the last three digits out on their own and test only those.
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