How would you test to see whether 7 is a factor of 291?

Answer 1

There is no trick to find out easily if 7 is a factor of a given number, like 291. There are easy rules for finding out if 3 is a factor of a given number (adding up the digits, again and again until you have a single digit and then see if the single digit is a multiple of 3 or not: 291 includes digits 2, 9, 1 2 + 9 + 1 = 12 1 + 2 = 3 3 is a multiple of 3, so 291 is a multiple of 3. Indeed 291 = 3 x 97).

For 7, the only was is to work out the division directly and see if there is any remainder or not: 291 / 7 = 41 remainder 4 thus 291 is not a multiple of 7, which is the same as saying that 7 is not a factor of 291.

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Actually there is a way to find whether a number has a factor of 7. It is discussed in Arthur Benjamin's "Secrets of Mental Math" Book, mental division, page 100. It states:

Add or substract multiples of 7 to leave the last digit position as zero, then eliminate and continue until nothing is left.

Example with 5292

5292 ends in 2, 6X7 42, substract 42

5292 -42 5250, eliminate the zero

525, ends in 5 7x5 35, substract525-35490, eliminate the zero.49, which we know is a multiple of 7 (7^2). > 5292 is a multiple of 7.

Another example

16184, ends in 4, substract 1416170>1617 ends in 7, you can also add up to zero. Add 631680 >168, ends in 8, substract 28140> 14 which is divisible by 7. This number has a factor of 7.

291 , last digit is one, 3x7 21, substract.291 - 21 250 > 25 which is not a multiple of 7. > 291 does not have a factor of 7.