Question : If the 7-digit number 678p37q is divisible by 75 and p is not a composite, then the values of p and q are:
Option 1: p = 5, q = 5
Option 2: p = 3, q = 0
Option 3: p = 3, q = 5
Option 4: p = 2, q = 5
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Correct Answer: p = 3, q = 5
Solution : 75 = 5 × 5 × 3 Since 7-digit number 678p37q is divisible by 75, 678p37q is divisible by 3 and 5 simultaneously. By the divisibility rule of 5, q should be 0 or 5. q = 0 or 5 -------------(i) By the divisibility rule of 3, 6 + 7 + 8 + p + 3 + 7 + q is a multiple of 3. ⇒ 31 + p + q = 33 or 36 or 39 or ...... When q = 0, p = 2 or 5 or 8 When q = 5, p = 0 or 3 Hence, the correct answer is p = 3, q = 5.
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