The number 3798125P369 is divisible by 7. What is the value of the digit P ?
- (a) 1
- (b) 6 ✓ UPSC's answer
- (c) 7
- (d) 9
Why the answer is (b)
• A quick test for 7 (valid because 1001 = 7 × 11 × 13): split the number into groups of three digits from the right and take the alternating sum; the number is divisible by 7 if that sum is.
• From the right, 3798125P369 splits into the groups 369, 25P, 981 and 37, i.e. 369, (250 + P), 981, 37.
• Alternating sum = 369 − (250 + P) + 981 − 37 = 1063 − P.
• 1063 − P must be a multiple of 7: 1063 = 7 × 151 + 6, so P = 6.
• Check: 37981256369 = 7 × 5425893767. Hence P = 6, option (b).
Why the other options are wrong
- (a) 1
- With P = 1 the alternating sum is 1062, which leaves remainder 5 on division by 7.
- (c) 7
- With P = 7 the alternating sum is 1056, which leaves remainder 6 on division by 7.
- (d) 9
- With P = 9 the alternating sum is 1054, which leaves remainder 4 on division by 7.
Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2021, held on 10 October 2021. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.