MCQ
If the number $7254 * 98$ is divisible by $22,$ then the digit at $*$ is:
- ✓$1$
- B$2$
- C$6$
- D$0$
✓
Answer
Correct option: A.
$1$
We know that, if a number is divisible by two coprime numbers, then it is divisible by their product also.
If $7254 * 98$ is divisible by $22,$ then it must be divisible by $2$ and $11$ both.
Since, the last digit is $8,$ so it is divisible by $2.$ Now, for divisibility by $11,$ the difference of sum of digits on even and odd places must be either zero or divisible by $11.$
Sum of digits at even places $= 9 + 4 + 2 = 15$
Sum of digits at odd places $= 8 + * + 5 + 7 = 20 + *$
Required difference $= (20 + *) - 15 = 5 + *$
For divisibility by $11, 5 + * = 11 * = 6$
Hence, the digit at $*$ is $6.$
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