Question
Wed September 05, 2012 By: Gayathri

The no.of 4 digited numbers formed from the digits {2,3,4,5,6} that are divisible by 6 is ?

Expert Reply
Wed September 05, 2012

A number will be divisible by 6, if the sum of its digits is divisible by 3 and the ones place digit is an even number.

Among 2, 3, 4, 5 and 6, the even numbers are 2, 4 and 6.

So, the ones place digit of the required number is 2, 4 or 6.

 

Consider that the ones place digit of the four-digit number is 2.

So, sum of the remaining places digits must be of the form 3x + 1, where x is a natural number.

Except the ones place the remaining digits may be either of the following:

223, 226, 232, 235, 244, 253, 256, 322, 325, 334, 343, 346, 352, 355, 364, 424, 433, 436, 442, 445, 454, 463, 466, 523, 526, 532, 535, 544, 553, 556, 562, 565, 622, 625, 634, 643, 646, 652, 655, 664.

In such cases 40 such numbers are formed.

 

Consider that the ones place digit of the four-digit number is 4.

So, sum of the remaining places digits must be of the form 3x + 2, where x is a natural number.

Except the ones place the remaining digits may be either of the following:

224, 233, 236, 242, 245, 254, 263, 266, 323, 326, 332, 335, 344, 353, 356, 362, 365, 422, 425, 434, 443, 446, 452, 455, 464, 524, 533, 536, 542, 545, 553, 556, 563, 566, 623, 626, 632, 635, 644, 653, 656, 662, 665.

In such cases 43 such numbers are formed.

 

Consider that the ones place digit of the four-digit number is 6.

So, sum of the remaining places digits must be of the form 3x, where x is a natural number.

Except the ones place the remaining digits may be either of the following:

222, 225, 234, 243, 246, 252, 255, 264, 324, 333, 336, 342, 345, 354, 363, 366, 423, 426, 432, 435, 444, 453, 456, 522, 525, 534, 543, 546, 552, 555, 564, 624, 633, 636, 642, 645, 654, 663, 666.

In such cases 39 such numbers are formed.

 

Therefore, total number of required 4-digit numbers formed is 40 + 43 + 39 = 122.

 

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