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Which of the following numbers is completely divisible by 88?
24376
24384
24392
24408
24376
Check the last 3 digits for divisibility by 8 and the alternating sum of digits for divisibility by 11. Often, checking divisibility by 8 first eliminates most options quickly.
A set of four numbers: 24376, 24384, 24392, 24408. We need to identify which number is completely divisible by 88.
NтЙб0(mod88)тЯ║NтЙб0(mod8)┬аand┬аNтЙб0(mod11)
Check the last 3 digits for divisibility by 8 and the alternating sum of digits for divisibility by 11. Often, checking divisibility by 8 first eliminates most options quickly.
Students often test only one rule (either 8 or 11) and assume the number is divisible by 88, forgetting that a number must satisfy both simultaneously.
Rule Application
A number is divisible by 88 if it is divisible by both 8 and 11. For 8, the last three digits must be divisible by 8. For 11, the difference between the sum of digits at odd places and even places must be a multiple of 11 (including 0).
88=8├Ч11
Testing Divisibility by 8
Checking the last 3 digits of each option: 376/8 = 47 (Yes), 384/8 = 48 (Yes), 392/8 = 49 (Yes), 408/8 = 51 (Yes). All options pass the rule for 8.
24376├╖8=3047,24384├╖8=3048,24392├╖8=3049,24408├╖8=3051
Testing Divisibility by 11
Checking the alternating sum of digits for 24376: (2+3+6)тИТ(4+7)=11тИТ11=0. Since 0 is divisible by 11, 24376 is divisible by 11. Testing 24384: (2+3+4)тИТ(4+8)=9тИТ12=тИТ3 (No).
тИг(2+3+6)тИТ(4+7)тИг=0
A is correct because 24376 satisfies the divisibility rules for both 8 and 11, making it divisible by 88.
Understanding divisibility rules is essential for simplifying fractions and solving HCF/LCM problems in competitive exams efficiently.