Number Properties Practice: Test of Divisibility

This TANCET MBA, TANCET MCA practice question is a quant question from Number Systems, Number Properties and Number Theory. Concept tested is divisibility of numbers by 3, 4, 5, and 7. Test of divisibility is a must know aptitude concept not only for TANCET but also for exams such the GMAT, GRE, CAT, and SAT.

Question

Easy Quant

Find the greatest five digit number that is exactly divisible by 7, 10, 15, 21 and 28.

99,840
99,900
99,960
99,990
99,970

Correct Answer: Choice (3). 99960

Explanatory Answer

Step 1: Prime factorize all the divisors to find the least number of tests to be done.
The number should be divisible by 10 (2, 5), 15 (3, 5), 21 (3, 7), and 28 (2², 7).
Hence, it is enough to check whether the number is divisibile by 3, 2², 5 and 7. i.e., for 3, 4, 7, and 7.

What are the tests of divisibility for 3, 4, 5, and 7?

Test of divisibility by 3: Sum of the digits will be divisible by 3 if a number is divisible by 3. For e.g., sum of the digits of 345 is 3 + 4 + 5 = 12. 12 is divisible by 3. So, 345 is also divisible by 3.
Test of divisibility by 4: The righmost two digits viz., the units and tens digits will be divisible by 4 if a number is divisible by 4. For e.g, 1232 is divisible by 4 because 32 is divisible by 4.
Test of divisibility by 5: The units digit is either 5 or 0.
Test of divisibility by 7: For up to 5 digit numbers, the quickest way is to actually divide the number by 7 and check out.

Step 2: Go from answer choices

Choice 1: 99840 : The number is divisible by 3, 4, and 5. But the number is not divisible by 7.

Choice 2: 99900 : The number is divisible by 3, 4, and 5. But the number is not divisible by 7.

Choice 3: 99960 : The number is divisible by all 3, 4, 5, and 7.

Level of difficulty : Easy

The correct answer is Choice (3).