Question

Which of the following numbers is divisible by 24
[a] 35718
[b] 63810
[c] 537804
[d] 3125736

Answer 1

Hint: Use the property that if gcd(m,n) =1 then m|a and n|a implies mn|a. Use this property taking m = 8 and n = 3 and use the divisibility tests of 8 and 3 to find which numbers are divisible by both 8 and 3. Then from the above theorem, those numbers will be divisible by 24.

Complete step-by-step answer:

Divisibility tests:
Divisibility by 2: If the number at units place is 0, 2, 4, 6 or 8 then the number is divisible by 2.
Divisibility by 3: If the sum of digits of the number is divisible by 3 then the number is divisible by 3.
Divisibility by 4: If the number obtained by taking the last two digits of the given number is divisible by 4 then the given number is divisible by 4.
Divisibility by 5: If the number at units place is 0 or 5, then the number is divisible by 5.
Divisibility by 7: Take the quotient obtained by dividing the number by 10 and subtract two times the digit at unit’s place from it. If the number so obtained is divisible by 7 then the original number is divisible by 7, e.g. Consider the number 1442.
The quotient obtained on dividing by 10 = 144 {Answer is the original number with unit’s place removed}. Two times units place = 4.
Subtracting we get 144-4 = 140. Since 140 is divisible by 7, 1442 is also divisible by 7.
Divisibility by 8: If the number obtained by taking the last two digits of the given number is divisible by 8 then the given number is divisible by 8.
We know that $24=8\times 3$ and gcd(8,3) = 1. Hence a number will be divisible by 24 if and only if it is divisible by both 8 and 3.
Option a: 35718.
Sum of digits = 3+5+7+1+8=24.
Since 24 is divisible by 3.
Since 18 is not divisible by 4, we have 35718 is not divisible by 4 and hence 35718 is not divisible by 8.s
Hence 35718 is not divisible by 24.
Option b: 63810
Sum of digits = 6 +3 +8 +1 +0 =18.
Since 18 is divisible by 3, 63810 is divisible 3
Since 10 is not divisible 4, 63810 is not divisible by 4, and hence 63810 is not divisible by 8
Option c: 537804.
Sum of digits = 5+ 3+7+8+0+4=27.
Since 27 is divisible by 3, 537804 is divisible by 3.
Since 804 is not divisible by 8, 537804 is not divisible by 8.
Hence 537804 is not divisible by 24.
Option d: 3125736
Sum of digits = 3 +1 +2 +5 +7 +3 + 6=27
Since 27 is divisible by 3, 3125736 is divisible by 3.
Also since 736 is divisible by 8, 3125736 is also divisible by 8.
Hence 3125736 is divisible by 24.
Hence option [d] is correct.

Note: [1] If gcd(m,n) =1 then m|a and n|a implies mn|a
Proof: It can be shown that if g is the gcd of a and b then there exist integers x and y such that
g = ax+by.
So if gcd(m,n) = 1, we have there exist integers x and y such that mx+ny = 1.
Multiplying both sides by a, we get
max+nay = a.
Since n|a, we have mn|max
Also, since m|a, we have mn|nay
Hence mn| max+nay
Hence mn|a.
Hence proved.