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UPSC Prelims

Mathematics

Theme:

Statistics / सांख्यिकी

Q5 2022, Statistics

An Identity Card has the number ABCDEFG, not necessarily in that order, where each letter represents a distinct digit (1, 2, 4, 5, 7, 8, 9 only). The number is divisible by 9. After deleting the first digit from the right, the resulting number is divisible by 6. After deleting two digits from the right of original number, the resulting number is divisible by 5. After deleting three digits from the right of original number, the resulting number is divisible by 4. After deleting four digits from the right of original number, the resulting number is divisible by 3. After deleting five digits from the right of original number, the resulting number is divisible by 2. Which of the following is a possible value for the sum of the middle three digits of the number?

किसी पहचान-पत्र की संख्या ABCDEFG है, किन्तु आवश्यक नहीं कि इसी क्रम में हो, जहाँ हर वर्ण किसी भिन्न अंक (केवल 1, 2, 4, 5, 7, 8, 9) को निरूपित करता है। यह संख्या 9 से विभाज्य है। दाहिने से पहला एक अंक मिटाने पर परिणामी संख्या 6 से विभाज्य है। मूल संख्या के दाहिने से दो अंकों को मिटाने पर परिणामी संख्या 5 से विभाज्य है। मूल संख्या के दाहिने से तीन अंकों को मिटाने पर परिणामी संख्या 4 से विभाज्य है। मूल संख्या के दाहिने से चार अंकों को मिटाने पर परिणामी संख्या 3 से विभाज्य है। मूल संख्या के दाहिने से पाँच अंकों को मिटाने पर परिणामी संख्या 2 से विभाज्य है। निम्नलिखित में से कौन-सा इस संख्या के बीचोबीच तीन अंकों के योग का संभव मान है?

Explanation Please wait..

The card number is ABCDEFG.

Case 1. It is given that number is divisible by 9 i.e. sum of the digits will be divisible by 9.

Thus numbers can be 1+2+ 4+5+7+8+9= 36 which is divisible by 9

Case 2. Deleting two numbers from right number is divisible by 5 means last number is 5 or 0.

Since there is no 0 among the digits as per 1^st step, E=5.

Case 3. Deleting one number from right resulting number is divisible by 6 i.e. numbers which are divisible by both 2 and 3 are divisible by 6.

This means, last digit of the given number is even and the sum of its digits is a multiple of 3.

We have three even numbers (2, 4 and 8) and these should be at places of F, B and D in any order. No other digit would be even.

Thus Possible value of CD are 12, 72 and 92.

Now We have to find C + D + E.

If CD = 12, C + D + E = 1 + 2 + 5 = 8,

if CD = 72, C + D + E = 7 + 2 + 5 = 14,

if CD = 92, C + D + E = 9 + 2 + 5 = 16.

Only matched option is a.

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