${53}^3-{46}^3-7^3$ is divisible by:

Important Questions on Number System

MEDIUM

CAT

IMPORTANT

Quantum CAT Also Useful for XAT>Chapter 1 - SNAP>CMAT>Q 91

A number of decimal system when written in base $\mathrm{n}, 2<\mathrm{n}<10$, we get a two digit number. Further if we reverse the digits of the obtained number in base 'n' we get a number which is twice of the original number in decimal system. The sum of original and resultant number both in decimal system for the largest possible value of n is:

EASY

CAT

IMPORTANT

Quantum CAT Also Useful for XAT>Chapter 1 - SNAP>CMAT>Q 92

A number of decimal system when written in base n, $2<n<10$, we get a two-digit number. Further, if we reverse the digits of the obtained number in base 'n', we get a number that is twice the original number in the decimal system. Then, how many values of n are possible?

MEDIUM

CAT

IMPORTANT

Quantum CAT Also Useful for XAT>Chapter 1 - SNAP>CMAT>Q 93

Pandavas won a hen in the war of the Mahabharat. They brought it on the $1\mathrm{st}$  January, $2002$. This hen gave birth to $7$ new hens on the very first day. After it every new hen irrespective of its age everyday gave birth (only once in a lifetime) to $7$ new hens. This process continued throughout the year, but no any hen had been died so far. On the $365\mathrm{th}$ day all the Pandavas shared equally all the hens among all the five brothers. The remaining (if these can not be shared equally) hens were donated to Krishna. The number of hens which the Krishna had received is:

EASY

CAT

IMPORTANT

Quantum CAT Also Useful for XAT>Chapter 1 - SNAP>CMAT>Q 94

Total number of natural numbers being the perfect square, whose square root is equal to the sum of the digits of the perfect square is:

EASY

CAT

IMPORTANT

Quantum CAT Also Useful for XAT>Chapter 1 - SNAP>CMAT>Q 95

At our training Institute the number of boys is same as that of the girls. Last week, except $\frac{2}{3}\mathrm{rd}$ of the girls all the students went to picnic, where they bought some samosas but later on they found exactly one dozen samosas were not fresh so those $12$ samosas had been thrown away. After it the samosas were divided equally between boys and girls. Further when boys dealtout the samosas equally among themselves $39$ samosas left undistributed, but when the girls dealtout the same number of samosas equally among themselves $12$ samosas were still left undistributed. The number of students at our training institute is:

EASY

CAT

IMPORTANT

Quantum CAT Also Useful for XAT>Chapter 1 - SNAP>CMAT>Q 96

Darwin Miya has $6$ kinds of fruits in large amount and has sufficient number of identical boxes to store the fruits. He can put at least $10$ and atmost $15$ fruits in any box and he put only one kind of fruits in a box. Further not more than $5$ boxes can contain same number of fruits. Maximum number of fruits that he can put in the boxes is:

EASY

CAT

IMPORTANT

Quantum CAT Also Useful for XAT>Chapter 1 - SNAP>CMAT>Q 97

In the above question if he is allowed to put the equal number of fruits in atmost $7$ boxes and he has only $33$ boxes and now he can put any kind of fruit with any other kind of fruits. At least how many boxes are there in which the number of fruits are same if he fills every box to its maximum capacity.

EASY

CAT

IMPORTANT

Quantum CAT Also Useful for XAT>Chapter 1 - SNAP>CMAT>Q 98

If $\mathrm{n}\in 1, 3, 5, 7,\dots$ etc., then the value of ${19}^{\mathrm{n}}-{23}^{\mathrm{n}}-{43}^{\mathrm{n}}+{47}^{\mathrm{n}}$ is necessarily divisible by: