Question

How many three-digit multiples of 6 have the sum of digits divisible by 6 if all digits are different?
a) 51
b) 52
c) 53
d) 54
e) 55​

Answer 1

Answer:

your correct answer is option (d) 54

Step-by-step explanation:

hope its help

Answer 2

Answer:

d) 54

Step-by-step explanation:

To Find:- Number of three-digit multiples of 6 that have the sum of digits divisible by 6 when all digits are different.

Solution:-

Let the 3-digit number divisible by 6 be XYZ.

Prime factorization of 6 = 3 × 2.

So we can say that a number will be divisible by 6 if it's divisible by both 3 and 2.

We know that the 3-digit multiples of 6 are 102, 108, 114, 120, 126, 132, 138, 144, 150, 156, 162, 168, 174, 180, 186, 192, 198, 204, 210, 216, 222, 228, 234, 240, 246, 252, 258, 264, 270, 276, 282, 288, 294, 300, 306, 312, 318, 324, 330, 336, 342, 348, 354, 360, 366, 372, 378, 384, 390, 396, 402, 408, 414, 420, 426, 432, 438, 444, 450, 456, 462, 468, 474, 480, 486, 492, 498, 504, 510, 516, 522, 528, 534, 540, 546, 552, 558, 564, 570, 576, 582, 588,  594, 600, 606, 612, 618, 624, 630, 636, 642, 648, 654, 660, 666, 672, 678, 684, 690, 696, 702, 708, 714, 720, 726, 732, 738, 744, 750, 756, 762, 768, 774, 780, 786, 792, 798, 804, 810, 816, 822, 828, 834, 840, 846, 852, 858, 864, 870, 876, 882, 888, 894, 900, 906, 912, 918, 924, 930, 936, 942, 948, 954, 960, 966, 972, 978, 984, 990, 996.

Acc. to the question, X + Y + Z must be divisible by 6.

Therefore, the number from the above list that sums up to a number that is divisible by 6 if all digits are different are:

132, 138, 150, 156, 174, 192, 198, 204, 240, 246, 264, 312, 318, 354, 372, 378, 390, 396, 402, 408, 420, 426, 462, 468, 480, 486, 510, 516, 534, 570, 576, 594, 624, 642, 648, 684, 714, 732, 738, 750, 756, 792, 798, 804, 840, 846, 864, 912, 918, 930, 936, 954, 972, 978.

Therefore, the above 54 numbers are the ones that re divisible by 6 and their sum is also divisible by 6 when all digits are different.

Therefore, option d) 54 is correct.

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