Mathematics
Using divisibility tests, determine which of the following numbers are divisible by 4, 6, 8, 9 or 11:
(i) 197244
(ii) 613440
(iii) 4100448
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Answer
(i) 197244
Divisibility by 4: Last two digits = 44, and 44 ÷ 4 = 11. Divisible by 4.
Divisibility by 6: Last digit 4 (even, divisible by 2). Sum of digits = 1 + 9 + 7 + 2 + 4 + 4 = 27 (divisible by 3). Divisible by 6.
Divisibility by 8: Last three digits = 244, and 244 ÷ 8 = 30.5. Not divisible by 8.
Divisibility by 9: Sum of digits = 27, divisible by 9. Divisible by 9.
Divisibility by 11: Sum at odd places (from right) = 4 + 2 + 9 = 15. Sum at even places = 4 + 7 + 1 = 12. Difference = 15 − 12 = 3, not divisible by 11. Not divisible by 11.
Hence, 197244 is divisible by 4, 6 and 9.
(ii) 613440
Divisibility by 4: Last two digits = 40, divisible by 4. Divisible by 4.
Divisibility by 6: Last digit 0 (divisible by 2). Sum of digits = 6 + 1 + 3 + 4 + 4 + 0 = 18 (divisible by 3). Divisible by 6.
Divisibility by 8: Last three digits = 440, and 440 ÷ 8 = 55. Divisible by 8.
Divisibility by 9: Sum of digits = 18, divisible by 9. Divisible by 9.
Divisibility by 11: Sum at odd places (from right) = 0 + 4 + 1 = 5. Sum at even places = 4 + 3 + 6 = 13. Difference = 13 − 5 = 8, not divisible by 11. Not divisible by 11.
Hence, 613440 is divisible by 4, 6, 8 and 9.
(iii) 4100448
Divisibility by 4: Last two digits = 48, divisible by 4. Divisible by 4.
Divisibility by 6: Last digit 8 (divisible by 2). Sum of digits = 4 + 1 + 0 + 0 + 4 + 4 + 8 = 21 (divisible by 3). Divisible by 6.
Divisibility by 8: Last three digits = 448, and 448 ÷ 8 = 56. Divisible by 8.
Divisibility by 9: Sum of digits = 21, not divisible by 9. Not divisible by 9.
Divisibility by 11: Sum at odd places (from right) = 8 + 4 + 0 + 4 = 16. Sum at even places = 4 + 0 + 1 = 5. Difference = 16 − 5 = 11, divisible by 11. Divisible by 11.
Hence, 4100448 is divisible by 4, 6, 8 and 11.
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